Affine Lie Algebras

Groups and Algebras for Theoretical Physics

Masters course in theoretical physics at The University of Bern Spring Term 2016

SUSANNE REFFERT

Contents

1 Complex semi-simple Lie Algebras 2 1.1 Basic notions ...... 2 1.2 The Cartan–Weyl basis ...... 3 1.3 The Killing form ...... 5 1.4 Weights ...... 6 1.5 Simple roots and the Cartan matrix ...... 7 1.6 The Chevalley basis ...... 9 1.7 Dynkin diagrams ...... 9 1.8 The Cartan classiﬁcation for ﬁnite-dimensional simple Lie algebras . . . . . 10 1.9 Fundamental weights and Dynkin labels ...... 12 1.10 The Weyl group ...... 14 1.11 Normalization convention ...... 16 1.12 Examples: rank 2 root systems and their symmetries ...... 17 1.13 Visualizing the root system of higher rank simple Lie algebras ...... 19 1.14 Lattices ...... 19 1.15 Highest weight representations ...... 22 1.16 Conjugate representations ...... 26 1.17 Remark about real Lie algebras ...... 27 1.18 Characteristic numbers of simple Lie algebras ...... 27 1.19 Relevance for theoretical physics ...... 27

2 Generalizations and extensions: Affine Lie algebras 30 2.1 From simple to affine Lie algebras ...... 30 2.2 The Killing form ...... 32 2.3 Simple roots, the Cartan matrix and Dynkin diagrams ...... 34 2.4 Classification of the affine Lie algebras ...... 35 2.5 A remark on twisted affine Lie algebras ...... 38 2.6 The Chevalley basis ...... 38 2.7 Fundamental weights ...... 39 2.8 The affine Weyl group ...... 41 2.9 Outer automorphisms ...... 45 2.10 Visualizing the root systems of affine Lie algebras ...... 47 2.11 Highest weight representations ...... 50

3 Advanced topics: Beyond afﬁne Lie algebras 55 3.1 The Virasoro algebra ...... 55 3.2 Lie superalgebras ...... 58 3.3 Quantum groups ...... 67

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Complex semi-simple Lie Algebras

Symmetries, and with them, groups and algebras are of paramount importance in theoretical physics. The basic concepts have already been introduced in the course Advanced Concepts in Theoretical Physics. This course will build on the material treated there, with a special emphasis on techniques that prove to be useful to the theorist. In order to be self-contained, a certain amount of repetition is however inevitable. This course consists of three parts. The first part is dedicated to simple Lie algebras, which are basically a theorist’s daily bread. The second part treats affine Lie algebras and the third generalizations beyond the affine case which keep appearing in various contexts in theoretical physics.1 While the topic is certainly mathematical, treating the structure theory of Lie algebras, this course is aimed at physicists. Proofs are generally not given and I do not work at the highest possible level of generality (I do not e.g. work over a general field F).

1.1 Basic notions

In the following, we will be working over the ﬁeld C, which simpliﬁes many things as it is algebraically closed. A Lie algebra g is a vector space equipped with an antisymmetric binary operation

[ , ] : g g g (1.1) × → which satisﬁes the Jacobi identity

[X, [Y, Z]] + [Z, [X, Y]] + [Y, [Z, X]] = 0, X, Y, Z g. (1.2) ∈ This binary operation is usually referred to as either a commutator or the Lie bracket. As you know from ACTP, a Lie algebra g describes the Lie group G in the vicinity of the identity via the exponential map

eiaX G for X g, (1.3) ∈ ∈ where a is a parameter. A representation associates to every element of g a linear operator on a vector space V which respects the commutation relations of the algebra. The maximal number of linearly independent states that generate V is the dimension of the representation. Relative to a

1In retrospect, I would have chosen a different title for the course, since it ended up being centered around simple Lie algebras and their generalizations while groups have not made an important appearance.

FS2016 2 Part 1. Complex semi-simple Lie Algebras given basis, each element of g can be represented as a square matrix, and the basis vectors are represented as column matrices. A representation is irreducible if the matrices representing the elements of g cannot all be brought into a block-diagonal form by a change of basis. A Lie algebra can be speciﬁed by a set of generators Ja and their commutation relations { } a b ab c [J , J ] = ∑ i fc J . (1.4) c ab The numbers fc are the so-called structure constants of the Lie algebra. The number of generators is the dimension of the Lie algebra. A simple Lie algebra is a Lie algebra that contains no proper ideal (no proper subset of generators La such that [La, Jb] La Jb). A semi-simple Lie algebra is a direct sum of simple Lie{ algebras.} In the following,∈ { } we ∀ will focus only on the simple and semi-simple cases.

1.2 The Cartan–Weyl basis

In many cases, we want to work independently from a specific basis. In the case of Lie algebras, however, choosing a particular basis is most convenient. In the following, we want to construct the generators of g in the standard Cartan–Weyl basis. While this choice is canonical in the case of finite-dimensional semi-simple Lie algebras, for more general cases, no fully canonical form exists. First we need to find the maximal set of commuting Hermitian generators Hi, i = 1, . . . , r, where r is the rank of the algebra g:

[Hi, Hj] = 0. (1.5)

This set of generators Hi forms the Cartan subalgebra h. The generators in h can be simultaneously diagonalized. The remaining generators Eα of g are chosen such that they satisfy

[Hi, Eα] = αiEα. (1.6)

The vector α = (α1,..., αr) is called a root2. The Eα are ladder operators. A basis satisfying both (1.5) and (1.6) is called a Cartan–Weyl basis (also standard or canonical basis). As h is the maximal Abelian subalgebra of g, the roots are non-degenerate. The root α naturally maps an element Hi h to the number αi: ∈ α(Hi) = αi. (1.7)

The roots are therefore elements of the dual of the Cartan subalgebra,

α h∗. (1.8) ∈ α † α With (E ) = E− , we see from hermitian conjugation of Eq. (1.6) that whenever α is a root, so is α. We use the notation − ∆ = set of all roots , (1.9) { } also called the root system. The root components αi can be regarded as the non-zero eigenvalues of the Hi in the adjoint representation, in which the Lie algebra g itself is

2The name is chosen as the αi are roots of the characteristic equation for Hi.

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the vector space on which the generators act. A matrix representation of the adjoint representation in the basis Ja is given by { } (Ja) = i f . (1.10) bc − abc In the adjoint representation, we have Eα Eα α , (1.11) 7→ | i ≡ | i Hi Hi , (1.12) 7→ | i identifying the generators and the states of the representation. The action of a generator X in the adjoint representation is ad(X)Y = [X, Y] (1.13) so that ad(Hi)Eα = αiEα Hi α = αi α . (1.14) 7→ | i | i The one-to-one correspondence between the α and the Eα reﬂects the fact that the roots | i are non-degenerate. In the adjoint representation, the zero eigenvalue has degeneracy r (associated to the Hi ). | i dim(adj. rep.) = dim(algebra) = r + # roots. (1.15) We now need to specify the remaining commutation relation of the algebra g in the Cartan–Weyl basis Hi, Eα . From the Jacobi identity, we ﬁnd { } [Hi, [Eα, Eβ]] = (αi + βi)[Eα, Eβ]. (1.16) Therefore,  [Eα, Eβ] ∝ Eα+β if α + β ∆,  ∈ [Eα, Eβ] = 0 if α + β / ∆, (1.17)  ∈ [Eα, Eβ] is a linear comb. of Hi if α = β. − We will use in the following the expressions r r α H = ∑ αi Hi, α 2 = ∑ αiαi. (1.18) · i=1 | | i=1 The full set of commutation relations of g in the Cartan–Weyl basis is given by [Hi, Hj] = 0, (1.19) [Hi, Eα] = αiEα (1.20)  N Eα+β if α + β ∆  α,β ∈ α β 2 if [E , E ] = α2 α H α = β (1.21)  | | · − 0 otherwise,

where Nα,β = const.

Example: The Cartan–Weyl basis for sl(2, C). The basis 0 1 1 0 0 0 E = , H = , F = (1.22) 0 0 0 1 1 0 − of sl(2, C) has commutation relations [H, E] = 2E, [H, F] = 2F, [E, F] = H. (1.23) − This fulﬁlls the commutation relations of a Cartan–Weyl basis.

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1.3 The Killing form

In order to define a scalar product on g, we will in the following define the Killing form. This will allow us, among other things, to fix normalizations. It is defined as follows:

Ke(X, Y) = Tr(adX adY), X, Y g. (1.24) ◦ ∈ The Killing form obeys X, Y g ∀ ∈ Ke(X, Y) = Ke(Y, X) symmetry, (1.25) Ke([X, Y], Z) = Ke(X, [Y, Z]) invariance. (1.26)

For semi-simple Lie algebras, the Killing form is non-degenerate:

Ke(X, Y) = 0 Y X = 0. (1.27) ∀ ⇒ This is an alternative way of deﬁning semi-simplicity.

Example: The Killing form of sl(2, C). Reusing the basis elements E, H, F in this ordering and their commutation relation from the last example, we ﬁrst express them in the adjoint representation:

0 2 0 2 0 0   0 0 0 − adE = 0 0 1 , adH = 0 0 0  , adF =  1 0 0 . (1.28) − 0 0 0 0 0 2 0 2 0 − Using these matrices, we can now directly calculate the Killing form on this basis:

Ke(E, E) = 0, Ke(E, H) = 0, Ke(E, F) = 4, (1.29) Ke(H, H) = 8, Ke(H, F) = 0, Ke(F, F) = 0. (1.30)

In matrix form, 0 0 4 Ke(X, Y) = 0 8 0 . (1.31) 4 0 0 F

Often, a normalized version of the Killing form is used instead: 1 K(X, Y) = Tr(adX adY), (1.32) 2g ◦ where g is the dual Coxeter number. The standard basis Ja is understood to be orthonor- { } mal with respect to K, K(Ja, Jb) = δa,b. (1.33) The same is true for the generators of the Cartan sub-algebra:

K(Hi, Hj) = δi,j. (1.34)

As the Killing form acts as a scalar product, it can be used to raise and lower indices:

a ad d b 1 f bc = ∑ f c[K(J , J )]− . (1.35) d

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a fabc is anti-symmetric in all three indices. In the orthonormal basis J , the position of the indices is irrelevant. { } From the cyclic property of the trace, it follows that

K([Z, X], Y) + K(X, [Z, Y]) = 0. (1.36)

The Killing form is uniquely characterized by this property. It also follows that

α α α α [E , E− ] = K(E , E− )α H, (1.37) · α α 2 K(E , E− ) = . (1.38) α 2 | | The fundamental role of the Killing form is to establish an isomorphism between the Cartan subalgebra h and its dual h∗: K(Hi, ) : h R, (1.39) · → for ﬁxed Hi. To every element γ h , there corresponds a Hγ h: ∈ ∗ ∈ γ(Hi) = K(Hi, Hγ). (1.40)

For a root α, we have in particular: Hα = α H = ∑ αi Hi. (1.41) · i

With this isomorphism, we ﬁnally also have a positive deﬁnite scalar product on h∗:

(γ, β) = K(Hβ, Hγ). (1.42)

Since roots are elements of h∗, this deﬁnes a scalar product on root space. In particular,

α 2 = (α, α). (1.43) | | 1.4 Weights

So far, we have studied the algebra g from the point of view of the adjoint representation which encodes the essential structure of the algebra. We have seen that in the adjoint, the eigenvalues of the Cartan generators are called the roots and that the Killing form induces a scalar product between them. We now study the more general context of a ﬁnite dimensional representation. For an arbitrary representation, we can always ﬁnd a basis λ such that {| i} Hi λ = λi λ . (1.44) | i | i The eigenvalues λi make up the vector

λ = (λ1,..., λr) (1.45)

which is called a weight. λ(Hi) = λi. (1.46) The scalar product between weights is ﬁxed by the Killing form. In the adjoint representation, weights are called roots. From [Hi, Eα] = αiEα, we learn that Eα changes the eigenvalue of a state by α:

HiEα λ = [Hi, Eα] λ + Eα Hi λ = (λi + αi)Eα λ . (1.47) | i | i | i | i

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If Eα is non-zero, it must be proportional to a state λ + α , therefore the name ladder or | i step operator for Eα. We are mostly interested in finite dimensional representations. For any λ in a finite- | i dimensional representation, there are p, q Z+ such that ∈ (Eα)p+1 λ ∝ Eα λ + pα = 0, (1.48) | i | i α q+1 α (E− ) λ ∝ E− λ qα = 0, (1.49) | i | − i for any root α. α α 2 Note that the triplet of generators E , E− , α H/ α form an su(2) subalgebra analogous to · | | + 3 + 3 3 J , J−, J : [J , J−] = 2J , [J , J±] = J±. (1.50) { } ± Due to this fact, many of the properties of simple Lie algebras can be analyzed to a large extent by making judicious use of the properties of su(2). If λ is in a finite-dimensional representation, also its projection onto su(2) is finite- dimensional.| i Let the dimension of the subalgebra su(2) be 2j + 1. From the state λ , the state with | i highest J3 projection m = j can be reached by a finite number, p, of applications of J+. q applications of J on the other hand lead to the state with m = j: − − (α, λ) (α, λ) j = + p, j = q. (1.51) α 2 − α 2 − | | | | Eliminating j leads to (α, λ) 2 = (p q). (1.52) α 2 − − | | Therefore, any weight in a finite-dimensional representation is such that (α,λ) is an integer. λ α 2 This is true in particular for λ = β, where β is a root. | |

1.5 Simple roots and the Cartan matrix

As we have seen, #roots = dim(g) rk(g) rk(g). (1.53) − The roots are therefore in general linearly dependent. Thanks to the Euclidean inner product on h∗, we can have a geometric interpretation of the roots ("root vector"). We can in particular introduce a hyperplane in root space, which itself must not contain any roots, and use it to split the root system into positive roots ∆+ which are on one side of the hyperplane, and negative roots ∆ on the other side. Note that this choice of hyperplane and therefore the splitting into positive− and negative roots is not unique and amounts to the choice of a particular basis. Since whenever α is a root, also α is a root, we have − ∆ = ∆+. (1.54) − − We can thus decompose the algebra as

g = h g+ g , (1.55) ⊕ ⊕ − α where g = spanC E± , α > 0 are the subalgebras spanned by the step operators for positive and± negative{ roots. This} is the so-called Gauss or triangle decomposition. A step

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operator Eα associated to a positive root α ∆ is called a raising operator, while E α ∈ + − with α ∆ is called a lowering operator. ∈ + A simple root αi is deﬁned to be a root that cannot be written as the sum of two positive roots. Geometrically speaking, the simple roots are those positive roots that are closest to the hyperplane used to separate positive and negative roots, where the notion of the norm is induced by the Killing form. There are exactly r = rk(g) simple roots:

∆ = α ,..., α , (1.56) s { 1 r} where the subscript i is a labeling index (not the index that refers to the root component!). The set of simple roots provides the most convenient basis for root space. Immediate consequences of the above deﬁnition are

• α α / ∆, i − j ∈ • any positive root is a sum of positive roots. The basis of simple roots is however in general not orthonormal. This fact is captured in the Cartan matrix, which is deﬁned as follows:

2(αi, αj) A = . (1.57) ij α 2 | j| We will see that the Cartan matrix summarizes the structure of a semisimple complex Lie algebra completely. Using Eq. (1.52), we see that the entries of the Cartan matrix are necessarily integers. Moreover, Aii = 2 and in general, Aij = Aji. The Cauchy–Schwarz inequality implies that 6 A A < 4, i = j. (1.58) ij ji 6 Thus, for i = j, the entries of the Cartan matrix are negative integers and can be equal 6 to 0, 1, 2, 3. For Aij = 0, at least one of Aij, Aji must be equal to 1. In the set of roots− of a− simple− Lie algebra6 at most two different lengths of roots are possible− (long and short). The ratio between the squared lengths of long and short roots can be either two or three. When all the roots have the same length, the algebra is called simply laced. For convenience, we introduce the notation

2αi α∨ = . (1.59) i α 2 | i|

αi∨ is called the coroot associated to the simple root αi. The scalar product between roots and coroots is always an integer. We can now write the Cartan matrix very compactly as

Aij = (αi, α∨j ). (1.60)

The highest root θ is the unique root for which ∑i mi is maximized in the expansion ∑i miαi. All elements of ∆ can be reached by repeated subtraction of simple roots from θ.

r r θ = ∑ aiαi = ∑ ai∨αi∨, ai, ai∨ N. (1.61) i=1 i=1 ∈

The coefﬁcients ai are called the marks or Kac labels, while the coefﬁcients ai∨ are called the comarks or dual Kac labels. Marks and comarks are related via 2 a = a∨ . (1.62) i i α 2 | i|

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The dual Coxeter number (which we have already used in the deﬁnition of the normalized Killing form Eq. (1.32)) is deﬁned as

r g = ∑ αi∨ + 1. (1.63) i=1

1.6 The Chevalley basis

We will see that the full set of roots can be reconstructed from the simple roots, which in turn can be straightforwardly extracted from the Cartan matrix. This fact is made manifest in the so-called Chevalley basis. In this basis, to each simple root correspond the three generators i α i α i 2αi H e = E i , f = E− i , h = · , (1.64) α 2 | i| with commutation relations

[hi, hj] = 0, [hi, ej] = A ej, [hi, f j] = A f j, [ei, f j] = δ hj. (1.65) ji − ji ij We see that all the structure constants in the Chevalley basis are integers. The remaining step operators are obtained by repeated commutations of the basic generators, subject to the Serre relations,

i 1 A j [ad(e )] − ji e = 0, (1.66) i 1 A j [ad( f )] − ji f = 0. (1.67)

These constraints encode the rules for reconstructing the full root system from the simple roots. Finally, i j K(h , h ) = (αi∨, α∨j ). (1.68) The fact that the Serre relations do not mix the generators ei and f i corresponds to the fact that the root system is separated into positive and negative roots. Since the commutation relations and the Serre relations can be expressed in terms of the Cartan matrix A, we see that A encodes all the information about the structure of the Lie algebra g. The abstract formulation of Lie algebras via the Cartan matrix is in fact a good starting point for generalizations.

1.7 Dynkin diagrams

All the information in the Cartan matrix can be encoded in a planar diagram, the so-called Dynkin diagram. To each Cartan matrix, a diagram made of vertices and connecting lines is associated:

• Each vertex in the diagram represents a simple root (therefore #vertices = rk(g)).

• Long roots are marked by , short roots by . ◦ •

• The vertices i and j are joined by Aij Aji lines, in particular: – Orthogonal roots correspond to disjoint vertices. – Vertices connected by one line correspond to simple roots spanning an angle of 2π 3 = 120◦.

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Figure 1.1: Root systems of the rank 2 Lie algebras

– Vertices connected by two lines correspond to simple roots spanning an angle of 3π 4 = 135◦. – Vertices connected by three lines correspond to simple roots spanning an angle 5π of 6 = 150◦. Dynkin diagrams in which vertices are joined only by a single line correspond to Lie algebras in which all simple roots have the same length, hence the name simply laced. Cartan matrices which differ only by a renumbering of roots lead to the same Dynkin diagram.

Example: Cartan matrices and Dynkin diagrams of rank 2 Lie algebras. This is the simplest non-trivial example, where we can see the power of the Cartan matrix. For rank two, the Cartan matrix is a 2 2 matrix. Based on its properties listed after the deﬁnition in Eq. (1.57), we have the following× inequivalent possibilities:

2 0 2 1 2 2 2 3 (1) : , (2) : − , (3) : − , (4) : − . (1.69) 0 2 1 2 1 2 1 2 − − − They correspond to the following Dynkin diagrams:

(1) : , (2) : , (3) : , (4) : (1.70)

(1) is actually not a simple Lie algebra (the Cartan matrix is block-diagonal), but semi- simple: it corresponds to A A . (2) corresponds to A , (3) to B and (4) to the 1 ⊕ 1 2 2 exceptional Lie algebra G2. The root diagrams are shown in Figure 1.1. We will meet these simple Lie algebras again soon.

1.8 The Cartan classiﬁcation for ﬁnite-dimensional simple Lie algebras

The enumeration of all possible Cartan matrices is purely combinatorial. It is possible to classify the ﬁnite-dimensional simple Lie algebras completely via their Cartan matrices, respectively their Dynkin diagrams. There are four inﬁnite series:

A (r 1), B (r 3), C (r 2), D (r 4), (1.71) r ≥ r ≥ r ≥ r ≥ plus ﬁve isolated cases:

E6, E7, E8, G2, F4, (1.72)

FS2016 10 Part 1. Complex semi-simple Lie Algebras where the subscript denotes the rank of the group. The algebras in the inﬁnite series are called the classical Lie algebras. They are isomorphic to the following matrix algebras:

Ar ∼= sl(r + 1), Br ∼= so(2r + 1), Cr ∼= sp(r), Dr ∼= so(2r). (1.73) The ﬁve isolated cases are called the exceptional Lie algebras. The restrictions on the ranks of the classical Lie algebras are imposed to avoid overcounting. Including all values of r leads to the following isomorphisms:

A = B = C = D , B = C , D = A A , D = A . (1.74) 1 ∼ 1 ∼ 1 ∼ 1 2 ∼ 2 2 ∼ 1 ⊕ 1 3 ∼ 3

The simple Lie algebras of the types Ar, Dr, E6, E7 and E8 are all simply laced. For Br, Cr and F4, the long roots are √2 times longer than the short roots, while for G2, the long root is √3 times longer than the short root. The Cartan matrices of the simple ﬁnite-dimensional Lie algebras are: Ar:  2 1 0 . . . 0 0  −  1 2 1 . . . 0 0   − −   0 1 2 1 . . 0 0  A =  − −  (1.75)  ......     0 0 0 . . 1 2 1  − − 0 0 0 . . 0 1 2 − Br:  2 1 0 . . . 0 0  −  1 2 1 . . . 0 0   − −   0 1 2 1 . . 0 0  A =  − −  (1.76)  ......     0 0 0 . . 1 2 2  − − 0 0 0 . . 0 1 2 − Cr:  2 1 0 . . . 0 0  −  1 2 1 . . . 0 0   − −   0 1 2 1 . . 0 0  A =  − −  (1.77)  ......     0 0 0 . . 1 2 1  − − 0 0 0 . . 0 2 2 − Dr:  2 1 0 . . . 0 0  −  1 2 1 . . . 0 0   − −   0 1 2 1 . . 0 0   − −  A =  ......  (1.78)    0 0 0 . . 2 1 1   − −   0 0 0 . . 1 2 0  − 0 0 0 . . 1 0 2 − E6:  2 1 0 0 0 0  −  1 2 1 0 0 0   − −   0 1 2 1 0 1  A =  − − −  (1.79)  0 0 1 2 1 0   − −   0 0 0 1 2 0  − 0 0 1 0 0 2 −

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E7:  2 1 0 0 0 0 0  −  1 2 1 0 0 0 0   − −   0 1 2 1 0 0 1   − − −  A =  0 0 1 2 1 0 0  (1.80)  − −   0 0 0 1 2 1 0   − −   0 0 0 0 1 2 0  − 0 0 1 0 0 0 2 − E8:  2 1 0 0 0 0 0 0  −  1 2 1 0 0 0 0 0   − −   0 1 2 1 0 0 0 0   − −   0 0 1 2 1 0 0 0  A =  − −  (1.81)  0 0 0 1 2 0 0 1   − −   0 0 0 0 1 2 1 0   − −   0 0 0 0 0 1 2 0  − 0 0 0 0 1 0 0 2 − F4:  2 1 0 0  −  1 2 2 0  A =  − −  (1.82)  0 1 2 1  − − 0 0 1 2 − G2: 2 3 A = − (1.83) 1 2 − Table 1.1 shows the Dynkin diagrams of the ﬁnite-dimensional simple Lie algebras.

1.9 Fundamental weights and Dynkin labels

Weights and roots live in the same r-dimensional vector space. The weights can be expanded in the basis of simple roots. However, for irreducible finite-dimensional representations, their coefficients are not integers. A more convenient basis is the dual of the simple coroot basis. It is denoted by ω and defined by the relation { i}

(ωi, α∨j ) = δij. (1.84)

The ωi are called the fundamental weights and the basis

ω , i = 1, . . . , r (1.85) { i }

is called the Dynkin basis. The components λi of a weight in the Dynkin basis are called the Dynkin labels,

r λ = ∑ λiωi λi = (λ, αi∨). (1.86) i=1 ⇔

The Dynkin labels of weights in ﬁnite-dimensional irreducible representations are always integers. Such weights are called integral. Whenever we write a weight in component

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Ar 1 2 3 r 1 r E6 − 1 2 3 4 5

Br 1 2 r 1 r E7 − 1 2 3 4 5 6

Cr 1 2 r 1 r E8 − 1 2 3 4 5 6 7

r 2 Dr F4 1 2 − 1 2 3 4 r 1 −

G2 1 2

Table 1.1: Dynkin diagrams of the ﬁnite-dimensional simple Lie algebras

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form λ = (λ1,..., λr), we understand these components to be Dynkin labels. Note that the elements of the Cartan matrix are the Dynkin labels of the simple roots:

αi = ∑ Aijωj, (1.87) j

i.e. the ith row of A is the set of Dynkin labels for the simple root αi. The Dynkin labels are the eigenvalues of the Chevalley generators of the Cartan subalgebra:

i i h λ = λ(h ) λ = (λ, α∨) λ , (1.88) | i | i i | i that is, hi λ = λ λ . (1.89) | i i| i Note the role of the position of the index:

i i i hi : eigenvalue of h (Dynkin label), λ : eigenvalue of H . (1.90) A weight of special importance is the one for which all Dynkin labels are equal to one:

ρ = ∑ ωi = (1, . . . , 1). (1.91) i This is the Weyl vector or principal vector and is alternatively deﬁned as

1 ρ = 2 ∑ α. (1.92) α ∆+ ∈ The scalar product of weights can be expressed in terms of a symmetric quadratic form matrix Fij: (ωi, ωj) = Fij. (1.93) F is the transformation matrix relating the Dynkin basis ω and the simple coroot basis ij { i} α : { i∨} ωi = ∑ Fijα∨j . (1.94) j The relation between the quadratic form and the Cartan matrix is given by

2 1 αj F = (A− ) | | . (1.95) ij ij 2

The scalar product between the weights λ = ∑ λiωi and µ = ∑ µiωi takes the form

(λ, µ) = ∑ λiµj(ωi, ωj) = ∑ λiµjFij. (1.96) i,j i,j

1.10 The Weyl group

The root system of a simple Lie algebra has a high degree of symmetry. Also, there are many equivalent choices for the basis of simple roots. The symmetries of the root system of a Lie algebra form a group, the automorphism group Aut(∆). A particularly interesting subgroup of Aut(∆) is the Weyl group W which we will study in detail in the following. A distinguished element of W can be inferred from the fact that if α is a root, so is α: the mapping − α α, α ∆ (1.97) 7→ − ∈ is clearly a symmetry of the root system.

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Example: Automorphisms of A1. For A1, there are only two roots, α and α, each of which can take the role of the simple root. Therefore, Eq. (1.97) is the only non-trivial− map. For any weight λ in a ﬁnite-dimensional representation of A1, also λ is a weight of that representation. This reﬂection therefore also leaves the weight system− invariant. Together with 1, Eq. (1.97) forms the group

Z = 1 , (1.98) 2 {± } hence Aut(∆A1 ) is isomorphic to Z2.

A map on the set of roots must fulﬁll the following two conditions in order to qualify as a symmetry of the root system:

• the map must be linear and invertible

• the map must be a permutation of the roots.

Therefore, Aut(∆) Sdim g r, the symmetric group with dim g r elements. In general, ⊂ − − the automorphism group is however much smaller than Sdim g r, as arbitrary permutations cannot be described by a linear map. − It is however not difficult to give elements of W explicitly. Consider the reflection sα with respect to the hyperplane (through 0) in root space perpendicular to a fixed root α:

(β, α) s β = β 2 α, (1.99) α − (α, α) i.e. we subtract from each root β ∆ twice the component in the α-direction. This is indeed a permutation of the roots. The∈ set of all such reﬂections with respect to roots forms the Weyl group W. The product of two Weyl reﬂections is given by the composition of maps (1.100) sαsα0 = sα0 sα.

Since sα is a reflection, it is its own inverse, and the unit element of the composition is the identity map. Note that the composition of reflections leads to reflections as well as rotations. A generic w W is therefore not of the form (1.99). ∈ By linearity, the action of the Weyl group extends naturally to the weight space of g:

s λ = λ (α∨, λ)α. (1.101) α − Since an arbitrary element w W is a product of reflections, it leaves the inner product invariant: ∈ (sαλ, sαµ) = (λ, µ). (1.102) Thus any w W is an isometry. The Weyl∈ group is generated by a small number of specific reflections of type (1.99), namely the reflections corresponding to the simple roots. These are the simple or fundamental Weyl reflections: s s , i = 1, . . . , r. (1.103) i ≡ αi Every w W can be written as a word (Weyl word) ∈

w = sisj ... sk (1.104)

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in the letters si. This decomposition is however not unique. The length l(w) of w is the minimum number of si among all possible decompositions of w. The signature of w is deﬁned as e(w) = ( 1)l(w). (1.105) − The si fulﬁll the relations 2 si = 1, sisj = sjsi, if Aij = 0. (1.106) These relations generalize to ( 2 if i = j mij (sisj) = 1, where mij = π (1.107) π θ if i = j, − ij 6

and θij is the angle between the simple roots αi and αj. Eq. (1.107) can be used as the defining relation of the Weyl group. Any group having such a representation is called a Coxeter group. On the simple roots, the action of si takes the simple form s α = α A α . (1.108) i j j − ji i We have seen, that W maps ∆ into itself. In consequence, it provides a simple way to generate the full root system ∆ from the simple roots by acting with all the elements of W on ∆s: ∆ = wα , wα ,..., wα w W . (1.109) { 1 2 r| ∈ } This makes it clear that any set w α for fixed w could serve as a basis of simple roots. { 0 i} 0 The Weyl group induced a natural splitting of the r–dimensional weight vector space into a fan of open cones Cw. These cones, whose number is equal to the order of W, are defined as C = λ (wλ, α ) 0, i = 1, . . . , r , w W (1.110) w { | i ≥ } ∈ and are called Weyl chambers. They intersect only at the reflecting hyperplanes, their boundaries (wλ, αi) = 0. One of the chambers (depending on the choice of simple roots) is distinguished: the unique chamber whose points have only positive Dynkin labels λ Z. i ∈ This is the fundamental or dominant Weyl chamber C0 (also called fundamental Weyl domain). A weight in this chamber is said to be dominant. The highest root θ is an example of a dominant weight. W acts transitively and freely on the Weyl chambers. When acting with W on C0 including its boundary, one obtains the whole root space. Conversely, for any weight λ / C , there exists a unique w W such that wλ C . The W orbit of a weight λ is given ∈ 0 ∈ ∈ 0 by wλ w W . The W orbit of every weight has exactly one point in C . { | ∈ } 0 We can label the Weyl chambers by the elements of W. The fundamental chamber C0 corresponds to the identity element of the Weyl group.

1.11 Normalization convention

Until now, all normalizations have been fixed with respect to the square lengths of the roots. To fully fix the notation, we need to give a specific value to these lengths. In the standard convention, the square length of the long roots is set equal to 2. We can fix our normalization completely by setting

θ 2 = 2, (1.111) | | since θ is necessarily a long root.

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1.12 Examples: rank 2 root systems and their symmetries

Example 1: A2. As we have seen, A2 has Cartan matrix 2 1 − (1.112) 1 2 − and two simple roots α1, α2 of the same length. Using Eq. (1.87), we ﬁnd α = (2, 1), α = ( 1, 2). (1.113) 1 − 2 − Making use of Eq. (1.95), we can write down its quadratic form matrix: 1 2 1 F = . (1.114) 3 1 2

The Weyl group of A2 contains for sure the elements 1, s1 and s2. We use Eq. (1.107) with 2π θ12 = 3 to ﬁnd the relation 3 (s1s2) = 1. (1.115)

From this, we find s1s2s1 = s2s1s2, meaning that no words with more than three elements can appear in W. The full Weyl group is thus given by W = 1, s , s , s s , s s , s s s . (1.116) { 1 2 1 2 2 1 1 2 1} Using the Weyl group, we can find the full root system of A2 by acting with the elements of W on the two simple roots. We find ∆ = α , α , α + α , α α , α , α . (1.117) { 1 2 1 2 − 1 − 2 − 1 − 2} The highest root is θ = α1 + α2. The roots are shown in Figure 1.2. The Weyl chambers and their labels are indicated in red.

Example 2: C2. As we have seen, C2 has Cartan matrix 2 1 − (1.118) 2 2 − and two simple roots α1, α2 of the different length. Using Eq. (1.87), we ﬁnd α = (2, 1), α = ( 2, 2). (1.119) 1 − 2 − Making use of Eq. (1.95), we can write down its quadratic form matrix: 1 1 1 F = . (1.120) 2 1 2

The Weyl group of A2 contains for sure the elements 1, s1 and s2. We use Eq. (1.107) with 3π θ12 = 4 to ﬁnd the relation 4 (s1s2) = 1. (1.121)

From this, we ﬁnd s1s2s1s2 = s2s1s2s1. The full Weyl group is thus given by W = 1, s , s , s s , s s , s s s , s s s , s s s s . (1.122) { 1 1 1 2 2 1 1 2 1 2 1 2 1 2 1 2} This time, we start from the highest root θ = 2α1 + α2 and construct the root system by repeated subtraction of the simple roots. We ﬁnd ∆ = 2α + α , α + α , α , α , α , α , α α , 2α α , . (1.123) { 1 2 1 2 1 2 − 1 − 2 − 1 − 2 − 1 − 2 } The root system is shown in Figure 1.3. The Weyl chambers and their labels are indicated in red.

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↵2 1 ↵1 + ↵2 = ✓

s s1s2 2 ↵ ↵ 1 1

s s s 1 2 1 s1s2 ↵1 ↵2 ↵ 2

Figure 1.2: Root system and Weyl chambers of A2

C2 s1 1 ↵2 ↵1 + ↵2 2↵1 + ↵2

s s 1 2 s2 ↵1 ↵ 1

s1s2s1 s2s1

↵2 2↵ ↵ ↵1 ↵2 1 2 s1s2s1s2 s2s1s2

Figure 1.3: Root system and Weyl chambers of C2

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Figure 1.4: Root systems of the rank 3 Lie algebras

1.13 Visualizing the root system of higher rank simple Lie algebras

The root systems of the rank two cases can be easily visualized via root diagrams in the plane. Also the rank three cases are still amenable to visualization, see e.g. Fig. 1.4. Higher dimensional cases however, must make recourse to some form of projection to two dimensions. The projection of choice for root systems is the one onto the Coxeter plane. A Coxeter element of a Lie group is the product of all simple Weyl reﬂections. Changing the order of the reﬂections leads to a conjugate Coxeter element. The Coxeter number h is the number of roots divided by the rank of the algebra. For a given Coxeter element w, there is a unique plane (the Coxeter plane) on which w acts by rotation by 2π/h. The Coxeter plane is used to draw diagrams of root systems: the vertices and edges roots are orthogonally projected onto the Coxeter plane, yielding a polygon with h-fold rotational symmetry. No root maps to zero, so the projections of orbits under w form h-fold circular arrangements and there is an empty center. The root systems of F4 (h = 12), E6 (h = 12), E7 (h = 18) and E8 (h = 30) in the Coxeter plane projection are displayed in Figures 1.5 to 1.8 (images sources: Wikimedia Commons).

1.14 Lattices

d Let (e1,..., ed) be a basis of d-dimensional Euclidean space R .A lattice is the set of all points whose expansion coefﬁcients (in terms of the speciﬁed basis) are all integers:

Ze + Ze + + Ze , (1.124) 1 2 ··· d i.e. the Z–span of e . There are three lattices that are of importance for Lie algebras: { i} • the weight lattice: P = Zω + Zω + + Zω , (1.125) 1 2 ··· r • the root lattice: Q = Zα + Zα + + Zα , (1.126) 1 2 ··· r • the coroot lattice: Q∨ = Zα∨ + Zα∨ + + Zα∨. (1.127) 1 2 ··· r

19 FS2016 Part 1. Complex semi-simple Lie Algebras

Figure 1.5: Root system of F4 in the Coxeter plane projection with 12-fold symmetry.

Figure 1.6: Root system of E6 in the Coxeter plane projection with 12-fold symmetry.

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Figure 1.7: Root system of E7 in the Coxeter plane projection with 18-fold symmetry.

Figure 1.8: Root system of E8 in the Coxeter plane projection with 30-fold symmetry.

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α2

ω2

ω1 s1s2 s2 α1

s1s2s1 s2s1

Figure 1.9: Root and weight lattices of A2.

Weights in finite-dimensional representations have integer Dynkin labels, hence they belong to P. The integers specifying the position of a weight in P are the eigenvalues of the Cheval- ley generators hi. The effect of the remaining generators of g is to shift the eigenvalues by an element of Q, the root lattice. Since roots are weights in a particular finite-dimensional representation, Q P. This means that by acting with Eα, a point in P is translated to ⊆ another point in P. For the algebras G2, F4 and E8, we have Q = P. In all other cases, Q is a proper subset of P and the quotient P/Q is a finite group. Its order P/Q is equal to det A. | | The distinct elements of the coset P/Q define the so-called congruence classes or conjugacy classes. A weight λ lies in exactly one congruence class. For any algebra g, the congruence classes take the form

r λ ν = ∑ λiνi mod P/Q (mod Z2 for g = D2l), (1.128) · i=1 | |

where the vector (ν1,..., νr) is called the congruence vector. Since the bases ω and α are dual, P and Q are dual lattices. A lattice is said to { i} { i∨} ∨ be self-dual if it is equal to its dual. For simple Lie algebras, only E8 has a self-dual weight lattice.

1.15 Highest weight representations

Lie algebras play their role in physics not as abstract algebras, but through their representations, which act on suitable representation spaces. The highest weight representations

FS2016 22 Part 1. Complex semi-simple Lie Algebras are a particularly interesting subclass. any finite-dimensional representation of a simple Lie algebra belongs to this class. As we will see, the key idea for the analysis of these representations is to reduce the problem to the representation theory of sl(2) which we have studied in Sec. 1.4. There we have seen that to each (simple) root, an sl(2) subalgebra α α i spanned by E i , E− i , H belongs. It follows that any representation space has a basis on which the whole Cartan subalgebra h acts diagonally. Any finite-dimensional irreducible representation has a unique highest weight state i λ , which is completely specified by its Dynkin labels λ(h ) = λi. Among all the weights in| i the representation, the highest weight is the one for which the sum of its coefficients is maximal when expanded in simple roots. Thus for any α > 0, λ + α cannot be a weight in the representation, so that Eα λ = 0, α > 0. (1.129) | i ∀ From Eq. (1.52) (in this case, p = 0), it is clear that the highest weight of a finite- dimensional representation is necessarily dominant. An irreducible finite-dimensional representation space of a semi-simple Lie algebra is characterized by the fact that it has a highest weight of multiplicity one and that this weight is dominant integral. Conversely, each dominant integral weight λ is the highest weight of a unique irreducible finite- dimensional representation Lλ. By abuse of notation, representations are often specified by their highest weight. The highest weight of the adjoint representation is θ.

Weights and multiplicities. Starting from the highest-weight state λ , all states in the | i representation space Lλ can be obtained by the action β γ η E− E− ... E− λ for β, γ,..., η ∆ . (1.130) | i ∈ + The set of eigenvalues of all states in Lλ is the weight system Ωλ. Any λ0 Ωλ is such that ∈ λ λ0 ∆ . (1.131) − ∈ + All the weights of a given representation lie therefore in exactly one congruence class. For the weight λ = λ ∑ n α , n Z+, we call ∑ n the level or depth. Whenever two 0 − i i i ∈ i weights λ, µ satisfy λ µ = β with β of the form β = ∑ n α , we say µ λ. This provides − i i ≤ a partial ordering of Ωλ. For any irreducible highest weight representation Lλ, there is a unique weight of depth 0, namely the highest weight λ. In order to ﬁnd all the weights λ Ω , we use again the representation theory of 0 ∈ λ sl(2), rewriting Eq. (1.52) as

(λ0, α∨) = λ0 = (p q ), p , q Z . (1.132) i i − i − i i i ∈ + λ0 is necessarily of the form

λ0 = λ n α , n Z . (1.133) − ∑ i i i ∈ + We now proceed level by level, using a simple algorithm which stems from the fact that all weights of a ﬁnite-dimensional irreducible sl(2)-representation are obtained from the highest weight by subtracting the positive root α:

• From the highest weight λ = ∑ λiωi, all weights of level 1 are obtained by subtracting those simple roots αi for which λi > 0.

• More generally, for an arbitrary weight λ0 of Lλ, when λi0 > 0, we can subtract the simple root αi λi0 times from λ0, producing weights that are all part of Ωλ.

• Proceeding recursively, we can produce in this way all distinct weights of Lλ until we arrive at a depth where all weights have negative Dynkin labels and the process terminates.

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(1, 1) ↵ ↵ 1 2

( 1, 2) (2, 1) ↵ ↵ 2 1 (0, 0) ↵ ↵2 1

( 2, 1) (1, 2)

↵ ↵1 2 ( 1, 1)

Figure 1.10: Weight diagram for the adjoint representation of A2.

Example: the adjoint representation of A2. The highest weight of the adjoint representation of A2 corresponds to the highest root, λ = (1, 1). As we have learned before, the two simple roots are α1 = (2, 1) and α2 = ( 1, 2). The weight diagram resulting from the above algorithm is given in− Figure 1.10. −

Example: the adjoint representation of C2. The highest weight of the adjoint representation of C2 corresponds again to the highest root, θ = 2α1 + α2. So λ = (2, 0). The two simple roots are α1 = (2, 1) and α2 = ( 2, 2). The weight diagram resulting from the above algorithm is given in− Figure 1.11. −

Example: a representation of D5. Let us study a representation of D5 with highest weight (1, 0, 0, 0, 0). The roots can be read off from the Cartan matrix given in Eq. (1.78). The weight diagram resulting from the above algorithm is given in Figure 1.12.

The above examples show that the multiplicity of a (non-highest) weight might be larger than one. In general, the "quantum numbers" λi do not characterize a weight vector completely. A complete speciﬁcation requires

1 (dim g r) = ∆+ (1.134) 2 − | |

1 labels, i.e. 2 (dim g 3r) quantum numbers in addition to the Dynkin labels. To compute these multiplicities,− we can use the Freudenthal recursion formula, which gives the

FS2016 24 Part 1. Complex semi-simple Lie Algebras

(2, 0) ↵ 1

(0, 1) ↵ ↵ 1 2

( 2, 2) (2, 1) ↵ ↵ 2 1 (0, 0) ↵ ↵2 1

( 2, 1) (2, 2)

↵ ↵1 2 (0, 1) ↵ 1

( 2, 0)

Figure 1.11: Weight diagram for the adjoint representation of C2.

(1, 0, 0, 0, 0) ↵ 1 ( 1, 1, 0, 0, 0) ↵ 2 (0, 1, 1, 0, 0) ↵ 3 (0, 0, 1, 1, 1) ↵ ↵ 4 5 (0, 0, 0, 1, 1) (0, 0, 0, 1, 1) ↵ ↵ 5 4 (0, 0, 1, 1, 1) ↵ 3 (0, 1, 1, 0, 0) ↵ 2 (1, 1, 0, 0, 0) ↵ 1 ( 1, 0, 0, 0, 0)

Figure 1.12: Weight diagram for the representation of D5 with highest weight (1, 0, 0, 0, 0).

25 FS2016 Part 1. Complex semi-simple Lie Algebras

multiplicities of the weight λ0 in the representation λ in terms of all the weights above it:

∞ 2 2 λ + ρ λ0 + ρ multλ(λ0) = 2 ∑ ∑ (λ0 + kα, α)multλ(λ0 + kα). (1.135) | | − | | α>0 k=1

Example: multiplicity of the weight (0, 0) in the adjoint representation of A2. We apply the recursion formula (1.135) to compute the multiplicity of (0, 0) from the last example. We know that k = 1 and that the multiplicities of all three weights above are each one. For the three positive roots, (λ0 + kα, α) = 2. Using λ = θ = ρ = α1 + α2, we ﬁnd

(8 2)mult (0, 0) = 2(2 + 2 + 2), (1.136) − θ

therefore multθ(0, 0) = 2.

Generally, the weights of the adjoint representation of any semi-simple Lie algebra g are the g–roots, each occurring with multiplicity one, and in additions the weight λ = 0 with multiplicity r. Note that all weights in a given W–orbit have the same multiplicity:

mult (wλ0) = mult (λ0) w W. (1.137) λ λ ∀ ∈

Finally, note that a ﬁnite-dimensional irreducible representation Lλ is always unitary.

i † i α † † (H ) = H , (E ) = E− (1.138)

results in the norm of any state λ L being positive deﬁnite. | i ∈ λ

1.16 Conjugate representations

In a ﬁnite-dimensional irreducible representation, there is obviously also a unique lowest weight state. It lies in the W–orbit of the highest weight λ, in the Weyl chamber exactly opposite to C0. This chamber corresponds to the longest element w0 of W. The lowest weight state is thus given by w0λ. The conjugate representation λ∗ has for its highest weight the negative of the lowest weight state λ∗ of Lλ:

λ∗ = (w λ). (1.139) − 0 The Weyl vector ρ is the highest weight of a self-conjugate representation:

ρ = (w ρ). (1.140) − 0 More generally, all the weights in are the negatives of those in . Ωλ∗ Ωλ The conjugation is related to the reflection symmetries of the Dynkin diagram. For Ar, the conjugation amounts to reversing the order of the finite Dynkin labels. A representation is self-conjugate if and only if its weight system is invariant under the change of sign. As for any root α, also α is a root, the adjoint representation of any Lie algebra is self- − conjugate. The Dynkin diagrams with reflection symmetry are Ar, Dr and E6. The operation of conjugation corresponds to this reflection, which is an automorphism of weight space (the nodes of the Dynkin diagrams corresponding to the fundamental weights). In order for

FS2016 26 Part 1. Complex semi-simple Lie Algebras

complex algebra compact form real form sl(n, C) su(n) sl(n, R) so(2l + 1, C) so(2l + 1) so(l + 1, l) sp(n, C) sp(n, 0) sp(n, R) so(2l, C) so(2l) so(l, l)

Table 1.2: Compact and normal real forms for the classical Lie algebras representations of these three cases to be self-conjugate, the Dynkin labels of the highest weights must be invariant under the reﬂections, i.e. they have to satisfy

Ar : λi = λr i+1, (1.141) − Dr (r even) : λr = λr 1, (1.142) − E6 : λ1 = λ5, λ2 = λ3. (1.143)

1.17 Remark about real Lie algebras

So far, we have always assumed that the base ﬁeld of g is C, making use of the fact that C is algebraically closed. This is not the case for R. This means in particular that the eigenvalue equation (1.6) which determines the roots and therefore encodes the abstract structure of a Lie algebra need not have a solution in R. The theory of real Lie algebras is therefore much more involved. It is however possible to construct real forms of the complex Lie algebras we have studied. A simple Lie algebra has several non-isomorphic real forms. There are two standard real forms one can construct for any complex simple Lie algebra, based on the observation that in a Cartan–Weyl basis, all structure constants are real. As a consequence, the real vector space spanned by all real linear combinations

i α ∑ ξi H + ∑ ξαE (1.144) i α ∆ ∈ is a real Lie algebra. This Lie algebra is the normal or split real form. It is the least compact real form. On the other hand, the compact real form is spanned with real coefﬁcients by iHi for the Cartan subalgebra and { } q q α α α α (i (α, α)/2)(˙E + E− ) ( (α, α)/2)(˙E E− ) (1.145) { ∪ − } for its orthogonal complement. All other real forms can be obtained from the compact form by multiplying suitable generators by a factor of i. Table 1.2 collects the compact and normal real forms for the classical Lie algebras.

1.18 Characteristic numbers of simple Lie algebras

In this section, we collect the characteristic numbers of the simple Lie algebras we have encountered, see Tables 1.3 and 1.4.

1.19 Relevance for theoretical physics

In the ﬁrst part of this course, we have studied the structure of abstract simple Lie algebras. In physics, Lie algebras and in particular their representations are omnipresent. In particle

27 FS2016 Part 1. Complex semi-simple Lie Algebras

Ar Br Cr Dr dimension dim(g) r2 + 2r 2r2 + r 2r2 + r 2r2 r dual Coxeter number g r + 1 2r 1 r + 1 2r −2 − − order of Weyl group W (r + 1)! 2rr! 2rr! 2(r 1)r! highest root |θ | (1, 0, . . . , 1)(0, 1, . . . , 0)(2, 0, . . . , 0)(0, 1,− . . . , 0) weight/root lattice P/Q Zr+1 Z2 Z2 Z4 r odd, Z2 Z2 r even congruence vector ν (1, 2, . . . , r)(0, . . . , 0, 1)(1, 2, . . . , r)(2, 4, . .× . , r 1, r) r odd (0, . . . , 0,− 1) r odd

Table 1.3: Characteristic numbers of the classical Lie algebras

E6 E7 E8 F4 G2 dimension dim(g) 78 133 248 52 14 dual Coxeter number g 12 18 30 9 4 order of Weyl group W 51840 2903040 696729600 1152 12 highest root θ| | (0, . . . , 0, 1)(1, 0, . . . , 0)(1, 0, . . . , 0)(1, 0, 0, 0)(1, 0) weight/root lattice P/Q Z3 Z2 1 1 1 congruence vector ν (1, 2, 0, 1, 2, 0)(0, 0, 0, 1, 0, 1, 1) ---

Table 1.4: Characteristic numbers of the exceptional Lie algebras

physics, we mostly deal with the A-series (e.g. when studying the gauge groups of the standard model, SU(3) SU(2) U(1)). For spin representations, orthogonal and special orthogonal Lie algebras× play a role.× Also in integrable systems such as spin chains, Lie algebras are of great importance. In the simplest model, the XXX spin chain, each lattice point carries a representation of su(2) (corresponding to a spin pointing either up or down), but in more general models, representations of any simple Lie algebra are admitted. In modern theoretical physics beyond the standard model, the whole machinery of simple Lie algebras is of paramount importance. Lie algebras show up in many contexts, such as grand unified gauge groups (e.g. SU(5), SO(10), SU(8), E6, O(16)), and global symmetries. In string theory, gauge groups of the A and D–series can be created via D–brane constructions in type II superstring theory, whereas the heterotic string has gauge group SO(32) or E E . 8 × 8 In quantum systems, however, classical symmetries do not carry over directly, and the concept of the central extension becomes necessary. Other generalizations of Lie algebras that lead to infinite-dimensional algebras also appear naturally in physical contexts. In integrable models, super algebras and infinite dimensional Yangian algebras appear. The second part of this course will therefore be dedicated to extensions and generalizations of simple Lie algebras.

Literature

This part of the course presents material that is contained both in [DMS97] and [FS97]. The notation of [DMS97] is used throughout, but some sections follow more closely the exposition of [FS97]. [DMS97] is sometimes a little terse, while [FS97] takes a more mathematical approach.

FS2016 28 Part 1. Complex semi-simple Lie Algebras

References

[DMS97] P. Di Francesco, P. Mathieu, and D. Sénéchal. Conformal ﬁeld theory. Graduate texts in contemporary physics. New York: Springer, 1997. ISBN: 0-387-94785-X. URL: http: //opac.inria.fr/record=b1119694. [FS97] J. Fuchs and C. Schweigert. Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists. Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press, 1997. ISBN: 978-0521541190.

29 FS2016 Part 2

Generalizations and extensions: Afﬁne Lie algebras

When considering quantum systems in physics, we often reach the limits of applicability of (semi-)simple Lie algebras. While at the level of classical mechanics or field theory, the symmetries of a physical systems are described by a Lie algebra g, in the quantum description of the same system, the Lie brackets are not recovered completely. Additional terms appear in the commutation relations (in physics, this phenomenon is referred to as a quantum anomaly). In order to describe the quantum theory in Lie algebraic terms, we must interpret these new constant terms as the eigenvalues of some new operators which have constant eigenvalues on any irreducible representation space of g. These new operators extend g to a closely related algebra g˜, via a so-called central extension. Not every given Lie algebra admits however a non-trivial central extension. The (semi-)simple Lie algebras we have studied so far, in particular, do not. Our aim in this part of the course is to construct (untwisted) affine Lie algebras gˆ. In order to do so, we need to extend our familiar simple Lie algebras g to an infinite- dimensional loop algebra which in turn receives a central extension and is supplemented by a derivation. We will associate to each finite-dimensional g an affine extension gˆ by adding an extra node related to the highest root θ to the Dynkin diagram of g. The introduction of this extra simple root will make the root system and the Weyl group of gˆ infinite dimensional. Also the highest weight representations become infinite dimensional, however, they can be organized in terms of a new parameter, the level. The discussion in Part 2 of this course will follow the one of Part 1 very closely, pointing out important differences to the finite-dimensional case.

2.1 From simple to afﬁne Lie algebras

Let us consider a generalization of g in which the elements of the algebra are also Laurent 1 polynomials in some variable t. The set of these polynomials is denoted by C[t, t− ]. The generalization 1 g˜ = g C[t, t− ] (2.1) ⊗ is called the loop algebra g˜. By a loop in a topological space , one means the smooth embedding of a circle into , together with a chosen parametrization.M The loop algebra M associated to g consists of the space of analytic mappings from the circle S1 to g via the map t = eiγ, γ R. (2.2) ∈

FS2016 30 Part 2. Generalizations and extensions: Afﬁne Lie algebras

The generators of g are given by Ja tn. The algebra multiplication rule is the natural ⊗ extension from g to g˜:

a n b m ab c n+m [J t , J t ] = ∑ i f c J t . (2.3) ⊗ ⊗ c ⊗ A central extension is obtained by adjoining to g˜ a central element: a n b m ab c n+m ˆ a b [J t , J t ] = ∑ i f c J t + k n K(J , J )δn+m,0, (2.4) ⊗ ⊗ c ⊗ where kˆ commutes with all Jas and K is the Killing form of g. Assuming again the generators Ja to be orthonormal with respect to the Killing form and using the notation

Ja Ja tn, (2.5) n ≡ ⊗ we can rewrite the commutation relations as a b ab c ˆ [Jn, Jm] = ∑ i f c Jn+m + k n δa,bδn+m,0, (2.6) c These relations must be supplemented by a ˆ [Jn, k] = 0. (2.7) While seemingly ad hoc, the central extension is actually unique, as we will see in the following. Let us start with the generic cocommutatormutator

l a b ab c ˆi ab [Jn, Jm] = ∑ i f c Jn+m + ∑ k (di )nm, (2.8) c i=1 containing l central terms. Except for when n + m = 0, the central terms can be eliminated by a redeﬁnition of the generators. So

a b ab c [J0, Jn] = ∑ i f c Jn, (2.9) c a a i.e. the generators Jn transform in the adjoint representation of of g (ad(J0 )). Since { } a the central extensions commute with all the generators Jn , they are invariant tensors of the adjoint representation. There is, however only one (up{ } to normalization) such tensor, namely the Killing form itself. Therefore, only one central element can be added to the loop extension of a simple Lie algebra. The only central extension of a simple Lie algebra compatible with the antisymmetry of the commutators and the Jacobi identity is indeed the one of Eq. (2.6). As in the simple case, we want to rewrite everything in the (afﬁne) Cartan–Weyl basis. With the non-zero Killing norms

i j α α 2 K(H , H ) = δ , K(E , E− ) = , (2.10) ij α 2 | | the commutation relations become i j ˆ i,j [Hn, Hm] = k n δ δn+m,0, (2.11) i α i α [Hn, Em] = α En+m (2.12)  α+β Nα,βEn+m if α + β ∆  ∈ α β 2 ˆ if (2.13) [En, Em] = α2 α Hn+m + k n δn+m,0 α = β  | | · − 0 otherwise.

31 FS2016 Part 2. Generalizations and extensions: Afﬁne Lie algebras

1 r ˆ The set of generators H0 ,..., H0, k is manifestly Abelian. In the adjoint representation, { i ˆ } α i the eigenvalues of ad(H0) and ad(k) on the generators E are, respectively, α and 0. Being 1 r α independent of n, the eigenvector (α ,..., α , 0) is the same for all Em, m = 0, . . . , ∞, i.e. it is inﬁnitely degenerate. H1,..., Hr, kˆ is therefore not a maximal Abelian subalgebra. { 0 0 } It must be augmented by a new grading operator L0, whose eigenvalues in the adjoint representation depend on n: d L = t . (2.14) 0 − dt In the mathematics literature, usually the operator D = L is used instead, it is called a − 0 derivation. The action of L0 on the generators is

ad(L )Ja tn = [L , Ja tn] = nJa tn [L , Ja] = n Ja. (2.15) 0 ⊗ 0 ⊗ − ⊗ ⇒ 0 n − n a The element L0 measures therefore the mode number n of the generators Jn, i.e. the degrees with respect to the gradation are given by the eigenvalues of L0. The maximal Cartan subalgebra is generated by

H1,..., Hr, kˆ, L . (2.16) { 0 0 0} The other generators, Eα for any n and Hi for n = 0, play the role of ladder operators. n n 6 With the addition of L0, the resulting algebra is denoted as gˆ:

gˆ = g˜ Ckˆ CL . (2.17) ⊕ ⊕ 0 It is referred to as an untwisted affine Lie algebra. The addition of L0 has made the Killing form on gˆ non-degenerate and allows thus for a non-degenerate inner product on gˆ. Having an infinite number of generators Ja , n Z, { n} ∈ it is clearly an infinite-dimensional algebra. From the perspective of gˆ, g is referred to as a the corresponding finite algebra, generated by the zero modes J0 . In the physics literature, affine Lie algebras are often referred{ to} as Kac–Moody algebras. However the name Kac–Moody refers to a more general construction.

Example: the Heisenberg algebra. An already familiar inﬁnite-dimensional example is the algebra generated by the modes of a free boson:

[an, am] = n δn+m,0. (2.18)

This is the so-called Heisenberg algebra. It is the afﬁne extension of the u(1) algebra generated by the element a0. Comparing to the commutation relations Eq. (2.6), the level appears to be 1, however, the central terms can be changed arbitrarily by rescaling the modes. For the case uˆ(1), the level has no meaning.

2.2 The Killing form

In parallel to Part 1 of this course, we want to equip gˆ with an inner product. This means that we need to extend the Killing form from g to gˆ. We use again the identity Eq. (1.36). With X, Y Ja , Z = L , we get ∈ { n} 0 a b K(Jn, Jm) = 0, unless n + m = 0. (2.19)

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For n + m = 0, the t-factors disappear, leaving us with the Killing form of g, implying

a b a,b K(Jn, Jm) = δ δn+m,0. (2.20) Note that the affine Killing form is still orthonormal with respect to the finite algebra indices. The choice X, Z Ja , Y = kˆ yields ∈ { n} a ˆ ˆ ˆ K(Jn, k) = 0, K(k, k) = 0, (2.21) whereas Y = L0 leads to K(Ja, L ) = 0, K(L , kˆ) = 1. (2.22) n 0 0 − The only unspecified norm is K(L0, L0), which by convention is chosen to be

K(L0, L0) = 0. (2.23) This arbitrariness is related to the possibility of redeﬁning

L L + akˆ, a constant, (2.24) 0 → 0 without affecting the algebra. This redefinition of L0 changes its Killing norm by 2a. As in the finite case, the Killing form leads to an isomorphism between the elements− of the Cartan subalgebra and those of its dual and defines a scalar product for the latter. Take a state that is a simultaneous eigenvector of all the generators of the Cartan subalgebra. The components of the vector λˆ are given by the eigenvalues of this state:

λˆ = (λˆ (H1), λˆ (H2),..., λˆ (Hr); λˆ (kˆ); λˆ ( L )). (2.25) 0 0 0 − 0

λˆ = (λ; kλ; nλ) is called an affine weight. The scalar product induced by the Killing form is (λˆ , µˆ) = (λ, µ) + kλnµ + kµnλ. (2.26) As for the simple Lie algebras, affine weights in the adjoint representation are called affine roots. Since kˆ commutes with all the generators of gˆ, its eigenvalues on the states of the adjoint representation are 0. Affine roots are therefore of the form

βˆ = (β; 0; n). (2.27)

The scalar product between affine roots is therefore the same as of their simple counterparts, ˆ α (β, αˆ ) = (β, α). The affine root associated to the generator En is αˆ = (α; 0; n), n Z, α ∆. (2.28) ∈ ∈ If we define δ = (0; 0; 1), (2.29) i then nδ is the root associated to Hn. In the following, we use the notation α (α; 0; 0), (2.30) ≡ so that we can write the roots in Eq. (2.28) as

αˆ = α + nδ. (2.31)

The full set of roots is

∆ˆ = α + nδ n Z, α ∆ nδ n Z, n = 0 . (2.32) { | ∈ ∈ } ∪ { | ∈ 6 }

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The root δ is unusual as it has zero length,

(δ, δ) = 0. (2.33)

For this reason, it is called an imaginary root. All the roots in nδ are imaginary, { } (nδ, mδ) = 0, n, m. (2.34) ∀ All the imaginary roots have multiplicity r. The other roots have multiplicity one and are called real.

2.3 Simple roots, the Cartan matrix and Dynkin diagrams

As the next step, we want to identify a basis of simple roots for the affine Lie algebra gˆ. This corresponds again to splitting the root space into positive and negative roots. In such a basis, the expansion coefficients of any root are either all positive or all negative. The basis must contain r + 1 elements, r of which are necessarily the finite simple roots αi, while the remaining simple root must be a linear combination involving δ. The proper choice for the latter is α ( θ; 0; 1) = θ + δ. (2.35) 0 ≡ − − The basis of simple roots is thus given by

α , i = 0, . . . , r. (2.36) { i} The set of positive roots is given by

∆ˆ = α + nδ n > 0, α ∆ α α ∆ . (2.37) + { | ∈ } ∪ { | ∈ +}

This makes sense as α + nδ = α + nα0 + nθ = nα0 + (n 1)θ + (θ + α) and the expansion coefficients in terms of finite simple roots of the last two factors− are necessarily non-negative. Note, that in the affine case, there is no highest root. This implies also that the adjoint representation is not a highest weight representation.

Example: the root system of Aˆ 1. The algebra Aˆ 1 has exactly two simple roots, namely the simple root α = θ of the simple algebra A1 and α0 = θ + δ = α + δ. Its root system is given by − − ∆ˆ = α + nδ n Z . (2.38) {± | ∈ }

Given a set of simple roots and a scalar product, we can now deﬁne the extended Cartan matrix as Ab = (α , α∨), 0 i, j r, (2.39) ij i j ≤ ≤ where the afﬁne coroots are given by

2 2 2 α∨ = (α; 0; n) = (α; 0; n) = (α∨; 0; n). (2.40) αˆ 2 α 2 α 2 | | | | | | As for simple roots, we omit the hat for simple coroots, such that

α∨ = α , α∨ = (α∨; 0; 0), i = 0. (2.41) 0 0 i i 6

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Compared to the ﬁnite Cartan matrix, Abij contains an extra row and column. The extra entries are easily calculated in terms of the marks deﬁned in Eq. (1.61). We have

2 (α , α∨) = θ = 2 (2.42) 0 0 | | and r (α0, α∨j ) = (θ, α∨j ) = ∑ ai(αi, α∨j ). (2.43) i=0

The zeroth mark is defined to be a0 = 1. Since the finite part of α0 is a long root, the zeroth comark is also 1: 2 α0 a∨ = | | = 1. (2.44) 0 2 By construction, the extended Cartan matrix satisfies

r r ∑ ai Abij = ∑ Abija∨j = 0. (2.45) i=0 i=0

The dual Coxeter number is given by

r g = ∑ ai∨. (2.46) i=0

Again, all the information contained in the extended Cartan matrices can be encoded in extended Dynkin diagrams. The Dynkin diagram of gˆ is obtained from that of g by the addition of an extra node associated to α0. This node is linked to the αi-nodes by Ab0i Abi0 lines.

2.4 Classiﬁcation of the afﬁne Lie algebras

Just as the simple Lie algebras, affine Lie algebras can be classified completely via their Cartan matrices. The linear dependence between the rows of the extended Cartan matrix means that it has one zero eigenvalue, a reflection of the semi-positive nature of the affine scalar product. The rank of an (r + 1) (r + 1) affine Cartan matrix is thus r. As we will see, the classification of the extended Cartan× matrices relies directly on the classification of the simple Lie algebras.

Example: The afﬁne Lie algebras with r = 1. For r = 1, the extended Cartan matrices are 2 2 matrices and the requirement of det Ab = 0 is enough to determine the off-diagonal elements:× Ab Ab22 Ab Ab = 4 Ab Ab = 0. (2.47) 11 − 12 21 − 12 21 This leads directly to the two possibilities

2 2 2 4 Ab = − , Ab = − . (2.48) 2 2 1 2 − − The corresponding Dynkin diagrams are

0 1 , 0 1 , (2.49)

35 FS2016 Part 2. Generalizations and extensions: Afﬁne Lie algebras

so differently to the simple case, two nodes can be joined by four lines.

For higher rank cases, we start from the observation that by deleting the ith row and column for any i 0, . . . , r from an extended Cartan matrix, we obtain a Cartan matrix of a (semi)-simple∈ Lie { algebra.} By deleting further rows and columns, we always remain in the class of semi-simple Lie algebras. In particular, any 2 2 matrix obtained by deleting × r 1 rows and their corresponding columns from an extended Cartan matrix Ab must be one− of the rank 2 Cartan matrices we have studied. This restricts the matrix elements for i = j to 6 Ab Ab = 0 or min Ab , Ab = 1, max Ab , Ab 3. (2.50) ij ji {| ij| | ji|} {| ij| | ji|} ≤

We see that the rank 1 cases with Ab12 Ab21 = 4 are unusual. Implementing the above constraints, the enumeration of all possible afﬁne Cartan matrices of a given rank can be done by straightforward combinatorics.

Example: r = 2 with the submatrix of G2. For illustration, we study the rank 2 case with a 2 2 submatrix corresponding to the Cartan matrix of G : × 2 2 p q  Ab = r 2 3 . (2.51) − s 1 2 − The determinant is given by

det Ab = 2 p(2r + 3s) q(r + 2s). (2.52) − − We find that det Ab = 0 is only fulfilled for two combinations of the integers p, q, r, s, namely p = r = 1, q = s = 0 and p = r = 0, q = s = 1 . (2.53) { − } { − } ˆ (1) The affine Lie algebras corresponding to these two solutions are denoted G2 = G2 and (3) G2 .

Tables 2.1 and 2.2 show all the possible extended Dynkin diagrams resulting from this classification. The Dynkin diagrams in Table 2.1 are all untwisted affine Lie algebras, also (1) (1) denoted by Ar , Br , etc. In the examples we have however seen, that we also produce additional Cartan matrices. These additional Dynkin diagram are given in Table 2.2 and belong to the class of twisted affine Lie algebras which we have not discussed so far. For the latter class, a variety of naming conventions exist in the literature. In total, we have seven infinite series of affine Lie algebras

( ) ( ) ( ) ( ) A 1 (r 2), B 1 (r 3), C 1 (r 2), D 1 (r 4), (2.54) r ≥ r ≥ r ≥ r ≥ ( ) ( ) ( ) B 2 (r 3), C 2 (r 2), B˜ 2 (r 0), (2.55) r ≥ r ≥ r ≥ and nine exceptional cases,

(1) (1) (1) (1) (1) (1) A1 , E6 , E7 , E8 , F4 , G2 , (2.56) (2) (2) (3) A1 , F4 , G2 . (2.57)

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0 6 Aˆ r Eˆ6 1 2 3 r 1 r − 1 2 3 4 5

7 0

Bˆr Eˆ7 2 3 r 1 r 0 1 2 3 4 5 6 1 −

ˆ ˆ Cr 0 1 2 r 1 r E8 − 0 1 2 3 4 5 6 7

0 r

Dˆ r r 2 Fˆ4 2 3 − 0 1 2 3 4 1 r 1 −

ˆ ˆ A1 0 1 G2 0 1 2

Table 2.1: Dynkin diagrams of the untwisted afﬁne Lie algebras

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(2) (2) (2) 0 1 2 A1 0 1 Br , Dr+1

˜ (2) (2) (2) (2) Br , A2r 0 1 2 Cr , A2r 1 2 3 − 1

(2) (2) (3) (3) F4 , E6 G2 , D4 1 2 3 4 0 1 2 0

Table 2.2: Dynkin diagrams of the twisted afﬁne Lie algebras

(1) (1) For many purposes, it is natural to regard A1 as the first element of the series Ar and (2) ˜ (2) ˜ (2) A1 as the first element of Br . Note, that in Br three lengths of simple roots occur, necessitating a new notation for the Dynkin diagram: we have, in decreasing length, the types of nodes . (2.58) Inspecting Tables 2.1 and 2.2, we find that indeed, by removing any node from an extended Dynkin diagram, we get the Dynkin diagram of a simple Lie algebra.

2.5 A remark on twisted afﬁne Lie algebras

We have only constructed the untwisted affine Lie algebras so far, which is where our main interest lies. Yet we have seen that the twisted cases arise naturally in the classification of affine Lie algebras via their Cartan matrices. The twisted cases can be constructed in a similar way to the untwisted ones, by giving up the requirement of single-valuedness of the map from the circle S1 to g in the construction of the loop algebra, and instead imposing twisted boundary conditions. We shall however not do this in this course.

2.6 The Chevalley basis

The commutation relations of the generators of the Chevalley basis given in Eq. (1.65) have the following afﬁne extension:

i j 4 [h , h ] = (α∨, α∨)knδ δ = knδ δ , (2.59) n m i j ij n+m,0 α 2 ij n+m,0 | i| i j j [hn, em] = Ajien+m, (2.60) j j [hi , f ] = A f , (2.61) n m − ji n+m j j 2 [ei , f ] = δ h + knδ δ , i, j = 1, . . . , r. (2.62) n m ij n+m α 2 ij n+m,0 | i|

FS2016 38 Part 2. Generalizations and extensions: Afﬁne Lie algebras

These relation however do not involve only the generators of the r + 1 simple roots of gˆ and are not expressed in terms of the Cartan matrix of gˆ. In order to construct a genuine Chevalley basis, we need to add the generators

0 θ 0 θ 0 ˆ e = E1− , f = E 1, h = k θ H0 (2.63) − − · to the set of ﬁnite generators ei and f i. e0 and f 0 are the raising and lowering operators for α0. From now on, we will omit the mode index 0 from the ﬁnite g Chevalley generators. The commutation relations for the generators associated to the simple roots of gˆ can be written as

[hi, hj] = 0, (2.64) i j j [h , e ] = Abjien+m, (2.65) j [hi, f j] = Ab f , (2.66) − ji n+m i j j [e , f ] = δijh , i, j = 0, . . . , r. (2.67)

These commutation relations have to be supplemented by the afﬁne Serre relations

i 1 Ab j [ad(e )] − ji e = 0, (2.68) i 1 Ab j [ad( f )] − ji f = 0, i = j. (2.69) 6 In this form, it is manifest that Ab encodes the whole structure of gˆ. Its inﬁnite-dimensional nature is however not apparent.

Example: the Serre relations and roots of Aˆ 1. Using the Cartan matrix of Aˆ 1, we ﬁnd 1 Ab = 1 Ab = 3, so − 01 − 10 [ad(e0)]3e1 = [e0, [e0, [e0, e1]]] = 0, (2.70) [ad(e1)]3e0 = [e1, [e1, [e1, e0]]] = 0. (2.71)

This corresponds to the fact, that 3α0 + α1 and α0 + 3α1 are not roots, while for example α0 + 2α1 is a root. Using the fact, that δ = α0 + α1, we can rewrite the roots. In particular, the root system given in Eq. (2.38) can be rewritten as

∆ˆ = nα + mα m n 1, n, m Z . (2.72) { 0 1 | | − | ≤ ∈ }

We see again, that unlike in the simple cases, the root system of Aˆ 1 contains inﬁnitely many roots.

2.7 Fundamental weights

As in the simple case, the fundamental weights ωˆ i , 0 i r are deﬁned to be the elements of the basis dual to the simple coroots. The{ fundamental} ≤ ≤ weights are assumed to be eigenstates of L0 with zero eigenvalue. For i = 0, the afﬁne fundamental weights are given by 6

ωˆ i = (ωi; ai∨; 0). (2.73)

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Their finite part makes them dual to the finite simple roots, while the kˆ eigenvalue is fixed by the condition (ωˆ , α∨) = 0, i = 0. (2.74) i 0 6 The zeroth fundamental weight must have zero scalar product with all finite αis and satisfy

(ωˆ 0, α0∨) = 1. (2.75)

Hence it must be ωˆ (0; 1; 0). (2.76) 0 ≡ It is called the basic fundamental weight. With

ω (ω ; 0; 0), (2.77) i ≡ i it follows that

ωˆ i = ai∨ωˆ 0 + ωi. (2.78) The scalar product between the fundamental weights is given by

(ωˆ , ωˆ ) = (ω , ω ) = F , i, j = 0, (2.79) i j i j ij 6 (ωˆ , ωˆ ) = (ωˆ , ωˆ ) = 0, i = 0, (2.80) 0 i 0 0 6

where Fij is the quadratic form matrix of g. Afﬁne weights can be expanded in terms of the afﬁne fundamental weights and δ as

r ˆ λ = ∑ λiωˆ i + lδ, l R. (2.81) i=0 ∈

ˆ Since each fundamental weight contributes to the k eigenvalue by a factor of ai∨, we deﬁne

r ˆ ˆ k λ(k) = ∑ ai∨λi. (2.82) ≡ i=0

k is called the level. This implies that the zeroth Dynkin label λ0 is related to the ﬁnite Dynkin labels λ , i = 1, . . . , r and the level by { i} r ˆ ˆ λ0 = λ(k) ∑ ai∨λi, (2.83) − i=1 i.e. λ = k (λ, θ). (2.84) 0 − Modulo a possible δ factor, the relation between λˆ and its ﬁnite counterpart is simply

λˆ = kωˆ 0 + λ. (2.85)

Note, that roots are weights at level zero. Afﬁne weights are generally given in terms of Dynkin labels of the form

λˆ = [λ0, λ1,..., λr], (2.86)

e.g. ωˆ 0 = [1, 0, . . . , 0], ωˆ 1 = [0, 1, 0, . . . , 0], ωˆ r = [0, . . . , 0, 1]. (2.87)

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Note, however, that this notation does not keep track of the eigenvalue of L0. The Dynkin labels of simple roots are given by the rows of the afﬁne Cartan matrix:

αi = [Abi0, Abi1,..., Abir]. (2.88)

The afﬁne Weyl vector is deﬁned as

r ρˆ = ∑ ωˆ = [1, 1, . . . , 1], (2.89) i=0 and ρˆ(kˆ) = g, (2.90) the dual Coxeter number. Note, that unlike the simple case, ρˆ cannot be written as 1/2 the sum of positive affine roots. As in the simple case, affine weights whose Dynkin labels are all non-negative integers play a special role, they are called dominant. This property is however level-dependent, as λ0 is fixed by k and λi, i = 0, see Eq. (2.84). The set of all dominant weights of level k is k 6 denoted by P+. The finite part of an affine dominant weight is itself a dominant weight,

λˆ Pk λ P . (2.91) ∈ + ⇒ ∈ + 2.8 The afﬁne Weyl group

The Weyl reflection with respect to the real affine root αˆ is defined in complete analogy to the case of the simple Lie algebra:

s λˆ = λˆ (λˆ , αˆ ∨)αˆ , (2.92) αˆ − and the set of all such reflections generates the affine Weyl group Wb . Just like the Weyl group of a simple Lie algebra, it acts on the weight space by linear maps. Many of its properties are analogous to the simple case, but we will see that important differences arise. Wb is generated by the reflections si with respect to the simple roots. Each of these elementary reflections permutes the set of positive roots. The new feature in the affine case are related to the existence of the imaginary roots. As (δ, αˆ ) = 0, the imaginary roots are unaffected by the affine Weyl reflections:

s δ = δ (δ, αˆ ∨)αˆ = δ. (2.93) αˆ − Thus any Weyl reflection acts on the set of imaginary roots nδ n = 0 as the identity map. { | 6 } For i = 1, . . . , r, the si act precisely as the simple Weyl reflections of g. We will see however, that the reflection s0 with respect to α0 acts differently, namely as a reflection supplemented by a translation. The affine Weyl group acts therefore on the weight space of g as an affine mapping – hence the name affine for the affine algebras. With λˆ = (λ; k; n) and αˆ = (α; 0; m), we find

2m s λˆ = (s (λ + kmα∨); k; n [(λ, α) + km] ). (2.94) αˆ α − α 2 | | To analyze the structure of Wb , we want to rewrite this as

ˆ m ˆ (2.95) sαˆ λ = sα(tα∨ ) λ,

41 FS2016 Part 2. Generalizations and extensions: Afﬁne Lie algebras

↵2 1 ↵1 + ↵2 = ✓

s1 k =3

s s s2 1 2 k =2 ↵ ↵ 1 1 k =1

s s s 1 2 1 s1s2 ↵1 ↵2 ↵ 2

Figure 2.1: Shifted hyperplanes for s0 in the case Aˆ 2.

with defined as tα∨ tα = s α+δsα = sαsα+δ, (2.96) ∨ − i.e. 2 2 2 t λˆ = (λ + kα∨; k; n + [ λ λ + kα∨ ]/k ). (2.97) α∨ | | − | | The action of on the finite part of ˆ corresponds thus to a translation by the coroot . tα∨ λ α∨ Since (2.98) (tα∨ )(tβ∨ ) = tα∨+β∨ , the set of all generates the coroot lattice . An affine Weyl reflection is thus a product tα∨ Q∨ of a finite Weyl reflection times a translation by an appropriate coroot. As the group of such translations is infinite, the affine Weyl group is infinite-dimensional. Q∨ is an invariant subgroup of Wb : 1 w(t )w− = t , w Wb . (2.99) α∨ wα∨ ∀ ∈ As Q∨ and W only have the identity in common, Wb is isomorphic to the semi-direct product of W and the Abelian group T of translations by k-multiples of elements of Q∨:

Wb ∼= W n T = W n kQ∨. (2.100) While the afﬁne Weyl group is independent of the level, its action on those weights which have a deﬁnite value k of the level depends in a non-trivial manner on k. Let us now study the action of s0 in Eq. (2.94):

s0λˆ = (λ + kθ (λ, θ)θ; k; n k + (λ, θ)) = sθt θ(λˆ ). (2.101) − − − With s = θ, the finite part of s λˆ is s λ + kθ. This mapping can be described as a θ − 0 θ reflection with respect to an appropriately shifted hyperplane, see the example of Aˆ 2 shown in Figure 2.1. Just as in the simple case, the affine Weyl chambers are defined as those open subsets of the vector space of affine weights which are obtained by removing all hyperplanes that are left invariant by some Weyl reflection,

Cˆw = λˆ (wλˆ , α ) 0, i = 0, 1, . . . , r , w Wb . (2.102) { | i ≥ } ∈

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α2

λ2

λ1 α1

Figure 2.2: Afﬁne Weyl chambers of Aˆ 2 at for the level k = 2.

Due to the presence of the subgoup T Wb , the Weyl chambers are now polytopes of finite volume rather than infinite cones. In⊂ particular, they contain a finite number of weights. They are also referred to as Weyl alcoves. Note that because of the level dependence of T ∼= kQ∨, the size of the chambers depends on the level. Figure 2.2 shows the Weyl chambers of Aˆ 2 at for the level k = 2. The fundamental or dominant chamber corresponds to the element w = 1. Weights in this chamber have all Dynkin labels positive:

r ˆ λ = ∑ λiωˆ i + lδ, l R, λi 0. (2.103) i=0 ∈ ≥

To any weight λˆ which does not lie on the boundary of some chamber, there is a unique ˆ Weyl transformation wλˆ associated such that wλˆ (λ) lies in the fundamental afﬁne Weyl chamber. The afﬁne Weyl group preserves the scalar product,

2m (s λˆ , s λˆ ) = (s (λ + kmα∨), s (λ + kmα∨)) + 2k(n [(λ, α) + km] ) αˆ αˆ α α − α 2 | | = (λ, λ) + 2kn = (λˆ , λˆ ). (2.104)

Thus, all weights in a given Weyl orbit have the same length. A Wb orbit contains inﬁnitely many weights and has a unique weight in he fundamental Weyl chamber.

43 FS2016 Part 2. Generalizations and extensions: Afﬁne Lie algebras

k =4 k =3 s1s0s1 s0s1 s1 1 s0 s1s0 k =2 k =1 k =0 !4 !3 !2 !1 !1 !2 !3 !4 ··· ···

Figure 2.3: Afﬁne Weyl chambers of Aˆ 1 at levels k = 0, . . . , 4.

Example: the Weyl group of Aˆ 1. The affine Weyl group of Aˆ 1 is generated by the reflections s0, s1. Their actions on a weight λˆ = [λ0, λ1] are given by s λˆ = λˆ λ α = [ λ , λ + 2λ ], (2.105) 0 − 0 0 − 0 1 0 s λˆ = λˆ λ α = [λ + 2λ , λ ]. (2.106) 1 − 1 1 0 1 − 1 Let the level of λˆ be k. Using λ0 = k λ1, we can rewrite the simple affine Weyl reflections as − s λˆ = [ k + λ , 2k λ ], (2.107) 0 − 1 − 1 s λˆ = [k + λ , λ ]. (2.108) 1 1 − 1 We find s s λˆ = [ k λ , 2k + λ ]. (2.109) 0 1 − − 1 1 ˆ This means that s0s1 translates the finite part of λ by 2kωˆ 1 = kα1 = kα1∨. This means that s s corresponds to the basic translation operator t . The structure of W is 0 1 α1∨ b n n Wb = (s0s ) , s (s0s ) n Z . (2.110) { 1 1 1 | ∈ } The translation s0s1 has no finite order. The above expressions for the actions of the group elements can be used together with Eq. (2.102) to determine the boundaries of the affine Weyl chambers, which are displayed in Figure 2.3. We see that the size of chambers increases with the level.

Example: the Weyl group of Aˆ 2. The affine Weyl group of Aˆ 2 is generated by the reflections s0, s1, s2. Their actions on a weight λˆ = [λ0, λ1, λ2] are given by s λˆ = [ λ , λ + λ , λ + λ ], (2.111) 0 − 0 0 1 0 2 s λˆ = [ λ + λ , λ , λ + λ ], (2.112) 1 − 0 1 − 1 1 2 s λˆ = [ λ + λ , λ + λ , λ ]. (2.113) 2 − 0 2 1 2 − 2 Using again λ = k λ λ , we find 0 − 1 − 2 t = s s s s , t = s s s s . (2.114) α1∨ 2 0 2 1 α2∨ 1 0 1 2

The Weyl chambers of Aˆ 2 change in size with the level. For k = 2, they are shown in Figure 2.2.

FS2016 44 Part 2. Generalizations and Extensions

Part 2. Generalizations and ExtensionsPart 2. Generalizations and Extensions

Part 2. Generalizations and Extensions Part 2. Generalizations and Extensions

0 6 Part 2. Generalizations and Extensions Aˆ r Eˆ6 1 2 3 r 1 r 1 2 3 4 5 0 0 0 0 7 0 0 0 6 6 ˆ ˆ ˆ ˆ Ar 0 Ar E6 60 E6 6 ˆˆ ˆ ˆ Aˆ BArErˆ E6 E7 r 1 2 3 r 1 r 6 1 2 2 3 3 r 1 r r 1 2 r3 1 4 5 10 21 32 34 54 5 6 1 2 3 r 1 r Part 2. Generalizations11 2 3 r and1 extensions:r Afﬁne Lie algebras1 2 3 4 5 1 2 3 4 5

Part 2. Generalizations and Extensions 0 7 7 0 0 7 7 8 Aˆ0 Bˆ 0 r 0 r 6 Bˆr Bˆr Eˆ7 Eˆ7 ˆ Aˆ r CBˆrˆ Eˆ06 1 2 r 1 r Eˆ7 E8 Br 2 3 r 1 r E7 2 03 1 r2 1r3r 4 5 6 0 1 2 3 4 5 6 2 3 r 1r 02 13 2 r3 1 4 5 6 0 1 2 3 4 5 6 11 2 3 r 1 r 1 0 1 2 3 4 5 6 7 1 1 1 2 3 4 5 Dˆ 2l+1 Part 2. Generalizations and Extensions ˆ Cr 0 7r 8 8 8 8 0 ˆ r 2 ˆ ˆ ˆ Cˆr r CDˆrrˆ Eˆ8 1 2 1 r r E8 EFˆ48 Cr ˆ 0 0 1 1 2 2 r r1 r1 E8 ˆ00 1 2 r r 1 0 1 2 3 4 Br E7 2 3 2 3 r 1 r 10 010 121 232 r34 3145 4 56 5 67 6 70 01 12 23 43 54 65 76 7 1 0 Dˆ 2l Dˆ 2l 0 r 0 0 r r ˆ 0 r ˆ A1 0 61 8 G2 Aˆ Eˆ ˆ r 2 ˆ r Dˆ r 2 6 DFrˆ F4 2 rDˆ r r 2 Dˆ r4 Fˆ4 r 2 Fˆ4 0 01 12 3 4 ˆ 2 3 ˆ 02 103 21 32 4 3 4 0 1 2 3 4 1 C2r 30 2r 1 13 2r r 1 r 1 E18 22 33 4 r5 1 1 1 r r1 1 1 r 1 0 1 2 3 4 5 6 7 0 Table 2.1: Dynkin diagrams of the untwisted afﬁne Lie algebras

ˆ ˆ Aˆ 1ˆ 7 Gˆ2 A1ˆ E06 1 ÊG27 0ˆ 1 ˆ 0 0 A1 00 1 6 r A1 0G2 1 G20 1 2 ˆ ˆ 0 1 2 Ar E6 0 1 2 0 1 2 ˆ Dˆ r 2 ˆ Fˆ Br r E7 4 0 1 2 3 4 1 2 3 r 1 r 2 3 1 2 2 3 13 r 4 5 r 0 1 Table2 2.1:3 Dynkin4 5 diagrams6 of the untwisted affine Lie algebras 1 1 Table 2.1: Dynkinr diagrams1 of the untwisted affine Lie algebras Table 2.1: Dynkin diagrams of the untwistedTable 2.1: affine Dynkin Lie algebras diagrams of the untwisted affine Lie algebras Figure 2.4: Outer automorphisms7 of the Dynkin diagrams of the untwisted affine Lie 0 ˆ ˆ algebrasA1 0 1 G2 8 ˆ ˆ 0 1 2 Br ˆ E7 ˆ 2 3 r 1 r Cr 2.90 Outer1 20 automorphisms1 r 21 r3 4 5 E68 1 FS2016 0 1 2 3 4 5 6 736 Let D(g) be the symmetryTable 2.1: group Dynkin of the diagramsg Dynkin of the diagram untwisted and D affine(gˆ) the Lie symmetry algebras group of the gˆ Dynkin diagram. These are the sets of symmetry transformations of the simple roots 8 that0 preserve the scalarr product and hence the Cartan matrix. Since the scalar product of the affine roots only depends onFS2016 their finite parts, it is enough to consider36 the finite Cˆ FS2016r Eˆ 36 r 0 1 2 r 1 Dˆ projection8 of the systemr of2 simple roots. AFˆ simple root is thus mapped into another simple r 4 0 1 2 3 4 FS2016root with2 the3 same0 1 mark2 and3 comark.4FS20165 6 367 36 1We define the groupr of1 outer automorphisms of the Dynkin diagram of as gˆ (gˆ) = D(gˆ)/D(g). (2.115) 0 r O ˆ ˆ A1 0 1 G2 Dˆ r r 2 This quotientFˆ4 makes sense as D(g) is a subgroup of D(gˆ). (gˆ) is thus the set of symmetry 2 3 0 1 2 3 4 0 1 O2 FS2016transformations of the Dynkin diagram of gˆ36that are not symmetry transformations of the 1 r 1 Dynkin diagram of g. The outer automorphisms of the Dynkin diagrams of the untwisted affine Lie algebras are given in Figure 2.4. Table 2.1: Dynkin diagrams of the untwisted affine Lie algebras In terms of the action of their generating element on an arbitrary weight λˆ = [λ0,..., λr], ˆ ˆ A1 0 1 the outerG automorphism2 groups of the affine Lie algebras are given in Table 2.3 (for the cases, where 0 (gˆ)1= 12). Note that the action of (gˆ) does not change the level since every fundamentalO weight6 is mapped into another fundamentalO weight of the same comark. k It is clear that the set of dominant weights P+ is mapped into itself. Thus, (gˆ) preserves Table 2.1: Dynkin diagramsthe offundamental the untwisted Weyl affine chamber. Lie algebras O

45 FS2016

FS2016 36

FS2016 36 Part 2. Generalizations and extensions: Afﬁne Lie algebras

g (gˆ) action of generators O Ar Zr+1 aλˆ = [λr, λ0,..., λr 2, λr 1] − − Br Z2 aλˆ = [λ1, λ0,..., λr 1, λr] − Cr Z2 aλˆ = [λr, λr 1,..., λ1, λ0] − D2l Z2 Z2 aλˆ = [λ1, λ0, λ2 ..., λr, λr 1] × − a˜λˆ = [λr, λr 1,..., λ1, λ0] − D2l+1 Z4 aλˆ = [λr 1, λr, λr 2,..., λ1, λ0] − − E6 Z3 aλˆ = [λ1, λ5, λ4, λ3, λ6, λ0, λ2] E7 Z2 aλˆ = [λ6, λ5, λ4, λ3, λ2, λ1, λ0, λ7]

Table 2.3: Action of the generating element of the outer automorphisms

Let A be a generic element of (gˆ). Its action on an afﬁne weight is given by O r ˆ Aλ = kAωˆ 0 + ∑ λi A(ωˆ i ai∨ω0), (2.116) i=1 −

where k is the level of λˆ . This follows directly from Eq. (2.83). The second term on the r.h.s. of Eq. (2.116) acts on the finite part of λˆ like an automorphism on the finite weight lattice that leaves the origin fixed. It is, in fact, an element wA of the finite Weyl group. The sum in Eq. (2.83) is the affine extension of w λ at level zero, w λˆ kωˆ . Therefore, A A − 0 Aλˆ = k(A 1)ωˆ + w λ. (2.117) − 0 A

In general, wA can be characterized as follows. Let wi be the longest element of W(i), the subgroup of the ﬁnite Weyl group generated by all s , j = i. Then, j 6

wA = wiw0 for i such that Aωˆ 0 = ωˆ i, (2.118)

where w0 is the longest element of W. Note, that outer automorphisms must preserve the commutation relations of the algebra.

Example: the outer automorphism group of Aˆ 1. For Aˆ 1, the only non-trivial outer automorphism is a := ωˆ ωˆ . (2.119) 0 ↔ 1 Since W = 1, s , w = s . Comparing { 1} a 1 a[λ , λ ] = [λ , λ ] = [λ , k λ ] (2.120) 0 1 1 0 1 − 1 to

a[λ , λ ] = k(a 1)ωˆ + s [λ , λ ] (2.121) 0 1 − 0 1 0 1 = k(ω ω ) + [λ + 2λ , λ ] = [λ , k λ ], 1 − 0 0 1 − 1 1 − 1 we ﬁnd that this is correct.

FS2016 46 Part 2. Generalizations and extensions: Afﬁne Lie algebras 3.1 Hasse diagrams 55

(1,1)

(1,0) (0,1)

1 2 Figure 3.2: Hasse diagram of the positive roots of A2. The numbers (m ,m )denotethe root vector. Figure 2.5: Hasse diagram for the roots of A2.

The first step is to distribute the simple roots evenly on a horizontal line around Example: the outer automorphism group of Aˆ 2. For Aˆ 2, the basic element a maps the origin. This is achieved by the following horizontal projection Px of the simple roots ↵i: a := ωˆ ωˆ ωˆ ωˆ . (2.122) 0 → i 1 1→ 12 → 0 Px(↵i)= xi, (3.3) n 1 2 ⌘ We find that i = 1 and the longest element of W(1) is s2. Using w0 = s2s1s2 = s1s2s1, we find where n is the rank of the Lie algebra. The horizontal position of a generic root i ↵ = m ↵i can now be defined as wa = s2s2s1s2 = s1s2. (2.123) i i We find indeed by direct calculationPx (↵ that)=m Px (↵i)=m xi. (3.4) Note that the explicit summation of the index i has been dropped. From now on, a[λ , λ , λ ] = k(a 1)ωˆ + s s [λ , λ , λ ] (2.124) any contracted0 index1 2 will be− summed0 over.1 2 We0 can1 formalize2 the above a bit by introducing a projection= vectork(ω ' thatω ) +satisfies [λ + 2λ + λ , λ λ , λ ] 1 − 0 0 2 1 − 1 − 2 1 = [λ2, k λ(1↵ 'λ)=2, λx1]. = [λ2, λ0, λ1]. (3.5) − −i| i Expanded in the basis of simple co-roots, the projection vector ' explicitly reads

F1 ij ' =(A ) xj ↵i_. (3.6) When we take its inner product with a generic root ↵, we see that it indeed gives us 2.10 Visualizing the root systemsi of affine Lie algebras the desired projection (3.4): (↵ ')=m xi. The complete projection P =(Px, Py) can then be written as | We have seen how higher rank root systems of simple Lie algebras can be visualized in Px(↵)=(↵ ') , (3.7a) Sec. 1.13. Affine root systems, being infinite, necessitate| other means of visualizations. A good means of visualization arePHassey(↵)=( diagrams↵ ⇢_) , . A Hasse diagram is a(3.7b) graph that | displays the ordering between the different elements of a set, in our case, the roots. We where the projection in the vertical coordinate y is just the height of the root. have seen that we can define an ordering on the roots by saying one root is larger than Note that the horizontal coordinate (3.3) of a simple root ↵i strongly depends on the other,its numberα β, ifi. their If the difference order of the is simple a positive roots root. is changed, Additionally, the Hasse we diagram need to changes introduce a ≥ cover relationshape too.: a The root bestα is looking said to diagrams cover β, areα producedβ, if there when is no the root orderingγ smaller of nodes than in α and bigger thanthe Dynkinβ. We diagram can now (and draw thus a the Hasse ordering diagram of simple using roots) the following matches the rules: connections between the nodes. See also Figure 3.3. • If α β, the vertical coordinate for β is less than that for α. ≥ • If α β there is a straight line connecting α and β. Of course, we can also use Hasse diagrams for finite root systems, resulting in a finite graph. As a first simple example, the roots of A2 are displayed in Figure 2.5. The lines joining the roots correspond to the Weyl reflections with respect to simple roots which turn the simple root below into the one above. In the figures, each color and angle of a line encodes a Weyl reflection, see e.g. in Fig. 2.6 for the roots of A4. We can also use a Hasse diagram to visualize the Serre construction, where all elements of the algebra are produced via commutators of the Chevalley generators, see Figure 2.7 for the Serre construction of A4. While the root systems of simple Lie algebras can be visualized by other means, the true power of the Hasse diagrams is their ability to depict the infinite affine root systems. Figure 2.8 shows the roots systems of Aˆ 1, Cˆ2, Dˆ 4, Aˆ 8, Dˆ 7 and Eˆ7. The graphs display the

47 FS2016 Part 2. Generalizations56 and extensions: Afﬁne Lie algebras Chapter 3 Visualizations

(1,1,1,1) (1,1,1,1)

(1,1,1,0) (0,1,1,1) (1,1,1,0) (1,1,0,1)

(0,1,1,0)

(1,1,0,0) (0,0,1,1) (1,1,0,0) (1,0,1,0) (0,1,0,1)

(1,0,0,0) (0,0,0,1) (1,0,0,0) (0,0,0,1)

(0,1,0,0) (0,0,1,0) (0,1,0,0) (0,0,1,0)

1 2 3 4 3 1 2 4

Figure 3.3:FigureTwo 2.6: Hasse Dynkin diagram diagrams for the roots (below) of A . and Hasse diagrams (above) of the same Lie 58 Chapter4 3 Visualizations algebra, A4. The ordering of nodes in the left Dynkin diagram, indicated with numbers below the nodes, is canonical. The ordering of nodes in the right Dynkin diagram does not match the connections between them, resulting in a ade ade ade ade Hasse1 diagram2 with3 crossing4 lines. (a) Legend

[e1,e2] [e2,e3] [e3,e4] The lines drawn in a Hasse diagram represent the Weyl reﬂections in the simple [e1, [e12,e3]]e2 e3[e2, [e43,e4]] roots. Say there is a root ↵ projected to the point (x, y). Then the root ↵ + ↵i (b) Positive Chevalley generators (c) Single commutators connected to it by the line of the fundamental Weyl reﬂection wi gets projected to

the point (x + xi,y+ 1). The[e1, [e2 line, [e3,e4]]] of a fundamental reﬂection is therefore drawn at an angle given by

[e1, [e2,e3]] [e2, [e3,e4]] 1 1 wi = tan . (3.8) [e1,e2] [e3,e4] xi

Because xi is unique for all i,then distinct fundamental reﬂections wi all are drawn

at di(d)↵erentDouble commutators angles, and reflections(e) Triple commutators in the same simple root are drawn parallel. To distinguish between them even further they will get drawn in di↵erent colors, ranging Figure 3.5: The Serre construction for A . 4 th fromFigure blue 2.7: Hasse (the diagram first fundamental for the Serre construction reflection) of A4. to red (the n ). 3.1.1 VisualizingThe Hasse the Serre diagram construction of the full root system is symmetric around the origin, because

FS2016 Hasse diagramsof the can Chevalley serve as a neat involution tool48 to visualize (2.21). the results It is of therefore the Serre con- customary to draw only the positive struction,roots the step-by-step in a Hasse construction diagram. of the full algebra from the Cartan matrix (see Example 2.2). One then has to interpret the points in the diagram not as roots, but as the generatorFollowing they belong the to. above Furthermore, procedure the lines can it then is straightforward, be interpreted though sometimes tedious, as the adjoint action of the respective positive Chevalley generators. Starting at the bottom,to the draw vertical a steps Hasse in the diagrams diagram then of represent any root the steps system. of the Serre Figure 3.4 displays for example the construction.Hasse diagrams of various root systems. Figure 3.5 displays the Serre construction for the Lie algebra A4. One starts out with just the positive Chevalley generators (Figure 3.5b). The ﬁrst step is to take all (single) commutators [ei,ej] of the positive Chevalley generators that are consistent with the Chevalley relations (2.16), the Serre relations (2.17), and the Jacobi identity (2.3), which results in Figure 3.5c. This procedure is then iterated (Figure 3.5d and 3.5e) until it no longer yields new generators. The analogy presented above is only valid up to a certain point. The Serre Part 2. Generalizations and extensions: Afﬁne Lie algebras

Figure 2.8: Hasse diagram for the roots systems of Aˆ 1, Cˆ2, Dˆ 4, Aˆ 8, Dˆ 7 and Eˆ7.

49 FS2016 Part 2. Generalizations and extensions: Afﬁne Lie algebras

symmetry structure of the various afﬁne Lie algebras and give a more intuitive idea of their structure. All illustrations in this section are reproduced with permission from [Nut10], where the visualization via Hasse diagrams is explained in more detail.

2.11 Highest weight representations

In contrast to the case of simple Lie algebras, all non-trivial representations of an affine Lie algebra are infinite-dimensional. Again, the most interesting representations are the highest weight representations, as they allow for an easy investigation of unitarity and have many other similarities to the finite-dimensional case. Highest weight representations are characterized by a unique highest state λˆ which is annihilated by the action of all ladder operators for positive roots, | i

α α i E λˆ = E± λˆ = H λˆ = 0, n > 0, α > 0. (2.125) 0 | i n | i n| i The eigenvalue of this state, λˆ , is the highest weight of the representation

Hi λˆ = λi λˆ , kˆ λˆ = k λˆ , L λˆ = 0, i = 0. (2.126) 0| i | i | i | i 0| i 6

Setting the L0 eigenvalue to zero is a matter of convention; a redeﬁnition of L0 would yield any desired value. In the Chevalley basis, the eigenvalues are the Dynkin labels:

hi λˆ = λ λˆ , i = 0, . . . , r. (2.127) 0| i i| i All states in this representation are generated by the action of the lowering operators on λˆ . Since kˆ commutes with all the generators, these states all have the same kˆ eigenvalue. | i From now on, kˆ will be identified with its eigenvalue k, the level. In most applications, k is fixed from the onset. The analogues of the irreducible highest weight representations of g are those representations whose projections onto the sl(2) subalgebra associated to any real root are finite. It is sufficient to concentrate on simple roots.

The weight system Ωλˆ is the set of all weights in the representation of the highest- ˆ ˆ weight state λ . An analysis parallel to the simple case shows that any weight λ0 Ωλˆ satisﬁes | i ∈ (λˆ 0, α∨) = (p q ), i = 0, . . . , r (2.128) i − i − i for some positive integers pi, qi. This implies that

λ0 Z, i = 0, . . . , r. (2.129) i ∈

For the highest weight λˆ , pi = 0 for all i, therefore

λ0 Z , i = 0, . . . , r. (2.130) i ∈ + This requires in particular that

λ = k (λ, θ) Z . (2.131) 0 − ∈ + Since (λ, θ) Z , this means that k must be a positive integer, bounded from below by ∈ + (λ, θ): k Z , k (λ, θ). (2.132) ∈ + ≥ A far-reaching consequence of this constraint is that for a ﬁxed value of k, there can only be a ﬁnite number of dominant highest-weight representations. For example, for k = 1, the

FS2016 50 Part 2. Generalizations and extensions: Afﬁne Lie algebras

only such representations are those with highest weight ωˆ i such that the corresponding simple root αi has unit comark. Since a0∨ = 1 for all gˆ, ωˆ 0 is always dominant. The level-1 representation with highest weight ωˆ 0 is called the basic representation. For Aˆ r, all comarks are one. There are thus r + 1 dominant highest-weight representations at level 1 whose highest weights are the ωˆ i, i = 1, . . . , r. In the following, we will use the notation gˆk for the algebra gˆ at level k.

Example: Dominant highest weight representations of Aˆ 2 at level 2. For Aˆ 2, all the comarks are one, so the set of all dominant highest-weight representations at level two is given by [2, 0, 0], [0, 2, 0], [0, 0, 2], [1, 1, 0], [1, 0, 1], [0, 1, 1]. (2.133)

Example: Dominant highest weight representations of Gˆ2 at level 2. For Gˆ2, the comarks are a0∨ = a2∨ = 1, a1∨ = 2, which leads to the set of all dominant highest-weight representations at level two being

[2, 0, 0], [0, 0, 2], [1, 0, 1], [0, 1, 0]. (2.134)

Representations that decompose into finite irreducible representations of sl(2) and can further be written as a direct sum of finite-dimensional highest-weight spaces are called integrable. The adjoint representation, although not a highest weight representation is integrable. The first requirement is obviously met, and the direct sum decomposition is the root space decomposition into finite roots and imaginary roots. Dominant highest-weight representations are also integrable. Moreover, if a † a i † i α † α (Jn) = J n, or (Hn) = Hn, (En) = E−n, (2.135) − − dominant highest-weight representations are easily checked to be unitary. For instance,

α ˆ 2 ˆ α α ˆ 2 ˆ ˆ E n λ = λ En− E n λ = [nk (α, λ)] λ λ 0 (2.136) | − | i| h | − | i α 2 − h | i ≥ | | since for n > 0, any α and λˆ dominant

nk (α, λ) = [k (λ, θ)] + (n 1)k + (θ α, λ) 0. (2.137) − − − − ≥ The condition for the simple case given in Eq. (1.52) is for dominant highest weights equivalent to the existence of the singular vectors

αi θ E λˆ = E− λˆ = 0 (2.138) 0 | i 1 | i and αi λi+1 ˆ θ k (λ,θ)+1 ˆ (E0− ) λ = (E 1) − λ = 0, i = 0. (2.139) | i − | i 6 In the Chevalley basis, these vectors read

ei λˆ = ( f i)λi+1 λˆ = 0, i = 0, . . . , r. (2.140) | i | i

51 FS2016 Part 2. Generalizations and extensions: Afﬁne Lie algebras

In sharp contrast to the simple Lie algebras, the representation space Lλˆ resulting from quotienting out these singular vectors is not ﬁnite dimensional. The imaginary root δ can be subtracted from any weight without leaving the representation:

If λˆ 0 Ω , then λˆ 0 nδ Ω n > 0. (2.141) ∈ λˆ − ∈ λˆ ∀ The source of the inﬁnity lies in the absence of a singular vector similar to Eq. (2.139), but involving δ.

We will now study how the various weights in Ωλˆ can be obtained. The algorithm we used for g also works for gˆ, just involving one additional Dynkin label. In the affine case, however, the algorithm never terminates. We define the grade to be the L eigenvalue shifted such that L λˆ = 0 for the highest 0 0| i state λˆ . At grade zero, the states are obtained from λˆ by applying the finite Lie algebra | i | i generators, as they do not change the L0 eigenvalue. The finite projection of all the weights at grade zero are all the weights in the g irreducible finite-dimensional representation with highest weight λ. The weights at grade one are obtained from those at grade zero that have λ0 > 0 by subtraction of α0, followed again by the subtraction of all possible finite simple roots. The analysis of the higher grades follows the same pattern. An important point is that the finite projections of the affine weights at a fixed value of the grade are organized into a direct sum of finite-dimensional weight spaces. This shows that dominant highest-weight representations are integrable. Finally, we must give the multiplicity of each weight. When the L0 eigenvalue is taken into account, the multiplicities of the weights are clearly finite. It can be calculated from a modified version of the Freudenthal recursion formula Eq. (1.135) which keeps track of the root multiplicities:

∞ ˆ 2 ˆ 2 ˆ ˆ ˆ λ + ρˆ λ0 + ρˆ multλˆ (λ0) = 2 ∑ mult(αˆ ) ∑ (λ0 + pαˆ , αˆ )multλˆ (λ0 + pαˆ ). (2.142) | | − | | αˆ >0 p=1

Recall that real roots have multiplicity one, while imaginary roots have multiplicity r. Using our convention for the L0 eigenvalue of the highest weight states, the scalar product of two afﬁne highest weights does not differ from its ﬁnite form:

(λˆ , µˆ) = (λ, µ) for λˆ (L0) = µˆ(L0) = 0. (2.143) Thus, with λˆ = (λ; k; 0) and ρˆ = (ρ; g; 0),

λˆ + ρˆ 2 = λ + ρ 2. (2.144) | | | | However, at grade m, λˆ = (λ; k; m) and 0 − 2 2 λˆ 0 + ρˆ = λ0 + ρ 2m(k + g). (2.145) | | | | − Multiplicity calculations using the Freudenthal recursion formula are rather involved. The constancy of the weight multiplicities along Wb -orbits greatly simpliﬁes the analysis. The generating function for such multiplicities is called a string function. Let µˆ be a weight in Ω such that µˆ + δ / Ω and denote the set of such weights as Ωmax. The multiplicity λˆ ∈ λˆ λˆ of the various weights in the string µˆ, µˆ δ, µˆ 2δ,... is given by the string function − − ∞ λˆ n σµˆ (q) = ∑ multλˆ (µˆ nδ)q . (2.146) n=0 − For more complicated representations, several string functions are required. The complete information about the multiplicities of all the weights in the representation is contained in

FS2016 52 Part 2. Generalizations and extensions: Afﬁne Lie algebras

[1, 0]1 ↵ˆ 0 ↵ˆ [ 1, 2] 1 [1, 0] [3, 2] 1 1 1

[ 1, 2] [1, 0] [3, 2] 1 2 1

[ 1, 2] [1, 0] [3, 2] 2 3 2

[ 3, 4] [ 1, 2] [1, 0] [3, 2] [5, 4] 1 3 5 3 1

Figure 2.9: Weights at the ﬁrst few grades of the basic representation of (Aˆ 2)1. The multiplicity is given in the subscript. The colors encode the orbits under the Weyl group. the set of string functions σλˆ (q) for all µˆ Ωmax. However, since weight multiplicities are µˆ ∈ λˆ constant along Weyl orbits, that is,

wλˆ λˆ σµˆ (q) = σµˆ (q), (2.147) it is sufﬁcient to know the string functions for those weights in Ωmax that are also dominant λˆ (recall that a Weyl orbit contains exactly one element in the fundamental chamber). We ˆ note further, that all the weights in Ωλˆ must also be in the same congruence class as λ. The number of independent string functions required to fully specify the representation of highest weight λˆ is thus equal to the number of integrable weights at level k that are in the same congruence class as λˆ . For example, in (Aˆ 1)2, there are three integrable weights, [2, 0], [0, 2], [1, 1]. The ﬁrst two are in the same conjugacy class, therefore two string functions are needed in this case.

Example: the basic representation of (Aˆ 2)1. Let us consider the basic representation of (Aˆ 2)1 with highest weight [1, 0]. Using the algorithm described above, it is easy to write down the weights in the first few grades, see Figure 2.9. The first weight with non-trivial multiplicity is [1, 0] at grade 2. With λˆ = (0; 1; 0) and λˆ = (0; 1; 2), we find λˆ + ρˆ 2 = 1 0 − | | 2 and λˆ + ρˆ 2 = 1 12. To calculate the r.h.s. of Eq. (2.142), we must consider all the | 0 | 2 − weights λˆ + pαˆ for p, αˆ > 0, up to grade zero. The positive roots of Aˆ are α , α + nδ, nδ 0 2 1 ± 1 for n > 0, they all have multiplicity one. Finally, we find that the r.h.s. of Eq. (2.142) is equal to 24, resulting in multiplicity 2 for [1, 0] at grade 2. We have seen that we can simplify the calculation of the multiplicities by taking into account the Weyl orbits. We have already studied the action of the Weyl group Aˆ 1 on the Dynkin labels in an earlier exercise. We start with the dominant weight [1, 0] at grade 0. We find the orbit s s s s [1, 0] 0 [ 1, 2] 1 [3, 2] 0 [ 3, 4] 1 [5, 4] (2.148) −→ − −→ − −→ − −→ − ··· It is important to take into account that s0 increases the L0 eigenvalue of the weight it acts on and thus the grade by λ0. Thus, the second weight in the above sequence is at grade 1

53 FS2016 Part 2. Generalizations and extensions: Afﬁne Lie algebras

and the fourth and fifth are at grade 4. s1 does not change the grade. In Figure 2.9, the Wb -orbits are color coded. We see that each of the weights [1, 0] = (0; 1; m) represents one orbit. The orbits are consistent with the multiplicities of the weights given− in the figure. We see that the first few coefficients of the string function [1,0] are given by . σ[1,0] 1, 1, 2, 3, 5, . . . This is the number p(n) of inequivalent decompositions of n into positive integers (number of integer partitions) for which a closed formula exists:

∞ ∞ ∞ [ ] 1 σ 1,0 (q) = mult (µˆ nδ)qn = p(n)qn = , (2.149) [1,0] ∑ λˆ − ∑ ∏ 1 qn n=0 n=0 n=1 − it is the inverse of the Euler function.

Literature

The discussion on affine Lie algebras follows again [DMS97] and [FS97], using the notation of [DMS97] and supplementing some material from [FS97]. [Fuc92] treats affine Lie algebras in more mathematical detail, in particular the twisted case. The discussion of the visualization of affine root systems via Hasse diagrams as well as the illustrations can be found in [Nut10].

References

[Nut10] T. Nutma. “Kac-Moody symmetries and gauged supergravity”. PhD thesis. Rijksuniver- siteit Groningen, Dec. 2010. URL: http://www.rug.nl/research/portal/files/ 14628607/15_thesis.pdf. [DMS97] P. Di Francesco, P. Mathieu, and D. Sénéchal. Conformal ﬁeld theory. Graduate texts in contemporary physics. New York: Springer, 1997. ISBN: 0-387-94785-X. URL: http: //opac.inria.fr/record=b1119694. [FS97] J. Fuchs and C. Schweigert. Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists. Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press, 1997. ISBN: 978-0521541190. [Fuc92] J. Fuchs. Afﬁne Lie Algebras and Quantum Groups: An Introduction, with Applications in Conformal Field Theory. 1st. Cambridge Monographs on Mathematical Physics. Cam- bridge University Press, 1992. ISBN: 0521415934.

FS2016 54 Part 3. Advanced topics: Beyond afﬁne Lie algebras

Part 3

Advanced topics: Beyond afﬁne Lie algebras

In this last part of the course, we are entering advanced territory of further extensions of simple Lie algebras beyond the affine case or in different directions. The motivation for these extensions comes again from physics, they are the mathematical answers to some necessities that arise naturally in the physics context. Due to the advanced nature of these topics, this part of the course will have more the character of an overview. We will first visit the Virasoro algebra which arises naturally in two-dimensional conformal field theory and is a further extension of the untwisted affine Lie algebras. Next, we will consider Lie super-algebras, which bring the physical concept of fermions to Lie algebras by the inclusion of fermionic Dynkin nodes. Applications of this structure can be found for example in the study of integrable spin chains (the tJ model). Lastly, we will have a look at so-called quantum groups that solve the problem of how to deform a Lie algebra by a new parameter q. The simplest example of a supergroup is motivated by the XXZ spin chain, where the spins on the chain are subject to an anisotropy in the form of an external magnetic field in the Z-direction. Another important example of a quantum group is the Yangian, which appears in the algebraic Bethe ansatz resp. quantum inverse scattering method to solve integrable spin chains.

3.1 The Virasoro algebra

In the second part of this course, we have constructed the untwisted affine Lie algebras by extending the simple Lie algebras by a central element kˆ and a derivation L0. An important extension of the affine Lie algebras arises when we introduce another− central element c and infinitely many generators Ln, n Z. Their commutation relations are given by ∈

c 2 [Ln, Lm] = (n m)Ln+m + n(n 1)δ + , (3.1) − 12 − n m,0 [Ln, L¯ m] = 0, (3.2) c 2 [L¯ n, L¯ m] = (n m)L¯ n+m + n(n 1)δ + , (3.3) − 12 − n m,0 [Ln, c] = 0. (3.4)

55 FS2016 Part 3. Advanced topics: Beyond afﬁne Lie algebras

These bracket relations deﬁne a Lie algebra, the so-called Virasoro algebra. The relations involving both the Ln and the generators of gˆ read

[L , Ja] = n Ja , (3.5) m n − n+m ˆ a ˆ [Ln, k] = [c, Jn] = [c, k] = 0. (3.6)

Comparing with the commutation relations of L0 in Eq. (2.15) in the afﬁne case, the naming convention for the derivation suddenly makes sense, as it is now identiﬁed with the Virasoro generator L . − 0 Let us state a few important facts about the Virasoro algebra Vir. Vir possesses a triangle decomposition, Vir+ Vir0 Vir (3.7) ⊕ ⊕ − into the subalgebras generated by the positive, zero , and negative modes

Vir+ = span Ln n > 0 , Vir0 = span L0, c , Vir = span Ln n < 0 . (3.8) { | } { } − { | }

Vir0 is maximal Abelian subalgebra. Owing to the existence of a triangular decomposition, the representation theory of the Virasoro algebra parallels to a large extent the one of afﬁne Lie algebras. There are, in particular, highest-weight representations. We will brieﬂy study the representations with respect to the Ln. The representations of their anti-holomorphic counterparts are constructed along the same lines. As we have seen, the holomorphic and anti-holomorphic components of the overall algebra decouple. Since no pairs of generators in (3.1) commute, we choose a single operator, L0, which will be diagonal in the representation space, which in this context is also called a Verma module. We denote the highest-weight state by h , with eigenvalue h of L : | i 0 L h = h h . (3.9) 0| i | i

Since [L0, Lm] = mLm, Lm for m > 0 is a lowering operator for h, and L m, m > 0 a raising operator. We− adopt the convention, −

L h = 0, n > 0. (3.10) n| i A basis for the other states of the representation, the descendant states, is obtained by applying the raising operators in all possible ways:

L k L k ... L k h , 1 k1 kn, (3.11) − 1 − 2 − n | i ≤ ≤ · · · ≤

where, by convention, the L k appear in increasing order of ki. This state is an eigenstate − i of L0 with eigenvalue h0 = h + k + k + + k = h + N, (3.12) 1 2 ··· n where N is the level of the state (which differs from what we called the level in the afﬁne case, as there, it referred to the eigenvalue of the operator kˆ).

The lowest levels of a Verma module. The first few levels of the representation with highest state h are spanned by the states given in Table 3.1. Looking at the number | i of distinct, linearly independent states at level N, we find again the coefficients of the series of number of partitions p(N) of the integer N that we have already encountered in Eq. (2.149).

FS2016 56 Part 3. Advanced topics: Beyond afﬁne Lie algebras

level number of states states 0 1 h | i 1 1 L 1 h 2 − | i 2 2 L 1 h , L 2 h 3 − | i − | i 3 3 L 1 h , L 1L 2 h , L 3 h 4 2 − | i − − | i − 2| i 4 5 L 1 h , L 1L 2 h , L 1L 3 h , L 2 h , L 4 h − | i − − | i − − | i − | i − | i

Table 3.1: The ﬁrst few levels of the Verma module generated by h | i

† Using Lm = L m, we deﬁne an inner product on the Verma module. The inner product − of two states L k L k ... L k h and L l L l ... L l h is − 1 − 2 − n | i − 1 − 2 − n | i

h Lk ... Lk L l ... L l h , (3.13) h | m 1 − 1 − n | i where the dual state h satisfies h | h L = 0, j < 0. (3.14) h | l Note that the inner product of two states vanishes unless they belong to the same level. In general, two eigenspaces of a Hermitian operator (here, L0) having different eigenvalues are orthogonal. Since c is a central generator, all vectors in the Verma module generated by h have | i the same eigenvalue of c, called the central charge. The Virasoro algebra is intimately linked with conformal transformations in 2d and therefore of paramount importance in the study of 2d conformal field theory (CFT), i.e. a quantum field theory which is covariant under conformal transformations. A conformal transformation of the coordinates in d spacetime dimensions is an invertible mapping x x which leaves the metric tensor g invariant up to a scale: → 0 µν

gµν0 (x0) = Λ(x)gµν(x). (3.15)

A conformal transformation is locally equivalent to a (pseudo-)rotation and a dilation. The name conformal comes from the fact that the conformal group preserves angles. In d = 2, conformal invariance takes a new meaning. The reason is that here, there is an inﬁnite variety of locally conformal transformations, namely the holomorphic mappings of the complex plane onto itself. Consider the coordinates (z0, z1) on the plane. The covariant metric tensor gµν transforms under a change of the coordinate system given by zµ wµ(x) as → ∂wµ ∂wν gµν gαβ. (3.16) → ∂zα ∂zβ

The condition that the above transformation is conformal is g0(w) ∝ g(z), in other words,

2 2 2 2 ∂w0 ∂w0 ∂w1 ∂w1 + = + , (3.17) ∂z0 ∂z1 ∂z0 ∂z1 ∂w0 ∂w1 ∂w0 ∂w1 + = 0. (3.18) ∂z0 ∂z0 ∂z1 ∂z1 This corresponds in turn to

∂w1 ∂w0 ∂w0 ∂w1 = and = , (3.19) ∂z0 ∂z1 ∂z0 − ∂z1

57 FS2016 Part 3. Advanced topics: Beyond afﬁne Lie algebras

which are the Cauchy–Riemann equation for holomorphic maps. The result of the above is that CFTs in 2d can be solved exactly, which is why they hold such an important place in theoretical physics. Any holomorphic inﬁnitesimal transformation can be expressed as

∞ n+1 z0 = z + e(z), e(z) = ∑ cnz (3.20) n= ∞ − for the complex variable z = z0 + iz1, where by hypothesis, the inﬁnitesimal mapping admits a Laurent expansion around z = 0. The effect of such a mapping on a spinless and dimensionless ﬁeld φ(z, z¯) living on the complex plane is

¯ ¯ δφ = e(z)∂φ e¯(z¯)∂φ = ∑[cnlnφ(z, z¯) + c¯nlnφ(z, z¯)], (3.21) − − n where we have introduced the generators

l = zn+1∂ , l¯ = z¯n+1∂ . (3.22) n − z n − z¯ These generators obey the following commutation relations:

[l , l ] = (n m)l , (3.23) n m − n+m [ln, l¯m] = 0, (3.24) [l¯ , l¯ ] = (n m)l¯ . (3.25) n m − n+m Thus the conformal algebra is the direct sum of two isomorphic algebras (generated by the ln and l¯n, respectively). It is also called the Witt algebra. Each of these two inﬁnite-dimensional algebras contains a subalgebra generated by l 1, l0 and l1. This is the subalgebra associated with the global conformal group. Indeed,− from Eq. (3.20) we see that l 1 = ∂z generates translations on the complex plane, l0 = z∂z generates scale − − 2 − transformations and rotations, and l1 = z ∂z generates special conformal transformations. In CFT, however, one needs unitarizable− representations of the symmetry algebra. However, just like in the case of the loop algebra, the Witt algebra does not have any non-trivial unitary representations. Therefore, we must again introduce a central extension. It can be shown on quite general grounds that it must just consist of a complex number on the right hand side of the bracket relation (the central charge). In physics language, we say that the conformal symmetry develops an anomaly.

3.2 Lie superalgebras

Superalgebras are related to the concept of particles of different statistics, i.e. fermions and bosons. They contain even and odd generators, the odd ones being of fermionic nature. Mathematically speaking, these algebras have a Z2–grading. Lie superalgebras present a different kind of generalization of complex semi-simple Lie algebras from the ones we have studied so far. While they share many of the key concepts of simple Lie algebras such as the Cartan subalgebra, roots, weights, the Cartan matrix and Dynkin diagrams, new features arise due to the possibility of vanishing Killing forms. The latter give rise to a host of pathological possibilities, which makes many definitions involved in the study of superalgebras more bulky as they must either encompass or exclude them. We only have time for a brief look at simple Lie superalgebras and, apart from the basic definitions, will focus on their classification. We will concentrate on one class of

FS2016 58 Part 3. Advanced topics: Beyond afﬁne Lie algebras complex simple Lie superalgebras, namely the basic classical ones, which are nearest to the simple case we are familiar with. They also hold the most interest from the point of view of mathematical physics. The Lie superalgebras are however not "the superalgebra" that appears in particle physics beyond the standard model, which is an extension of the Poincaré algebra. Of course, all superalgebras share the basic Z2–grading. In the context of theoretical physics, Lie superalgebras appear as symmetry algebras of integrable spin chains, the simplest example being the tJ model.

First of all, we need to introduce the grading. Inspired by the properties of integer numbers, we introduce the following structure with product rules

even even = even, (3.26) · even odd = odd, (3.27) · odd odd = even. (3.28) ·

The above grading is also called a Z2–grading. Let us ﬁrst deﬁne a graded vector space V. Let V be a complex vector space of dimension m + n, m, n Z , and let A1,..., Am+n ∈ + be a basis of V. Then any A V can be written as ∈ m+n j A = ∑ aj A , aj C. (3.29) j=1 ∈

V can be graded by saying that

m j A = ∑ aj A , is even and (3.30) j=1 m+n j A = ∑ aj A , is odd. (3.31) j=m+1

Thus the even elements only involve the ﬁrst m basis elements, while the odd elements involve only the remaining n basis elements. Any A V that is either even or odd is said to be homogeneous, and the degree (or parity) of such∈ elements is deﬁned as ( 0 if A is even deg A = (3.32) 1 if A is odd.

The set of even elements of V forms the even subspace V0, the odd elements form the odd subspace V1: V = V V . (3.33) 0 ⊕ 1 If we supplement the graded vector space V by an associative product, the resulting structure is an associative superalgebra. Let gs be a graded vector space with g0 and g1 being its even, respectively odd subspaces and dim g = m, dim g = n, m 0, n 0, m + n 1. (3.34) 0 1 ≥ ≥ ≥ Assume that A, B g , there exists a generalized Lie product or supercommutator ∀ ∈ s [A, B] with the following properties:

• [A, B] g A, B g , ∈ s ∀ ∈ s

59 FS2016 Part 3. Advanced topics: Beyond afﬁne Lie algebras

• A, B g , a, b C, ∀ ∈ s ∀ ∈ [aA + bB, C] = a[A, C] + b[B, C], (3.35)

• for any two homogeneous A, B g , also [A, B] is homogeneous with degree ∈ s deg([A, B]) = (deg A + deg B) mod 2. (3.36)

• for any two homogeneous A, B g , ∈ s [B, A] = ( 1)(deg A)(deg B)[A, B], (3.37) − − • for any three homogeneous A, B, C g , ∈ s [A, [B, C]]( 1)(deg A)(deg C) + [B, [C, A]]( 1)(deg B)(deg A) − − + [C, [A, B]]( 1)(deg C)(deg B) = 0. (3.38) − (generalized Jacobi identity)

Then gs is called a complex Lie superalgebra with even dimension m and odd dimension 1 m+n n. We choose a homogeneous basis of gs in which the basis elements A ,..., A are 1 m m+1 m+n pq such that A ,..., A g0 and A ,..., A g1. Then, the structure constants cr can be deﬁned by ∈ ∈ m+n p q pq r [A , A ] = ∑ cr A . (3.39) r=1 Any generalized Lie product can be evaluated from the knowledge of the structure constants. The grading implies that pq p q qp c = ( 1)(deg A )(deg A )c . (3.40) r − − r For m 1, the even subspace g is an ordinary Lie algebra. For m 1, n 1, the ≥ 0 ≥ ≥ odd subspace g1 of gs is a carrier space for a representation of the Lie algebra g0. This representation is called the representation of g0 on g1. Every Lie algebra can be regarded as a special case of a Lie superalgebra for m 1, n = 0. ≥ When represented as matrices, the elements of a superalgebra are block matrices

AB M = . (3.41) CD

If M g , it has the form ∈ 0 A 0 M = , (3.42) 0 D while if M g , it has the form ∈ 1 0 B M = . (3.43) C 0 As in the simple case, we deﬁne ad(X)Y = [X, Y], but now using the supercommutator. The Killing form is deﬁned as

K(X, Y) = str(adX adY), X, Y g , (3.44) ◦ ∈ s where str is the supertrace, deﬁned as

strM = Tr A Tr D (3.45) −

FS2016 60 Part 3. Advanced topics: Beyond afﬁne Lie algebras for the block matrix M. A Lie superalgebra is said to be Abelian or commutative if

[A, B] = 0 A, B g . (3.46) ∀ ∈ s

A subalgebra g gs is a subset of elements of gs that form a vector subspace of gs and that is closed under⊂ the generalized Lie product,

[A, B] g for A, B g. (3.47) ∈ ∈ A graded subalgebra g g is itself a Lie superalgebra and its even subspace g is a s0 ⊂ s 0 subspace of g0 and its odd subspace g10 is a subspace of g1. gs0 is said to be a proper graded subalgebra of gs if at least one element of gs is not contained in gs0 . A graded subalgebra gs0 is invariant if [A, B] g0 A g0 , B g . (3.48) ∈ s ∀ ∈ s ∈ s A Lie superalgebra is said to be simple if it is not Abelian and does not possess a proper invariant graded subalgebra. If gs is a simple Lie superalgebra, its Killing form K( , ) is either non-degenerate or identically zero. We see that the situation is very different from the case of Lie algebras, where a Lie algebra with a zero Killing form cannot be semi-simple. Unlike in the case of Lie algebras, for Lie superalgebras, there do exist semi-simple Lie superalgebras that are not expressible as the direct sum of simple Lie superalgebras. In order to define them correctly, we first need to introduce the concept of solvability. Let (0) (k) (k 1) (k 1) gs = gs and gs = [gs − , gs − ] for each k = 1, 2, 3, . . . . Then the Lie superalgebra gs is ( ) solvable if there exists a value of k for which g k = 0 . s { } A Lie superalgebra gs is said to be semi-simple if it does not possess a solvable invariant graded subalgebra. Just as for the complex finite-dimensional simple Lie algebras, we want to classify the simple Lie superalgebras. We will however only look at the so-called classical ones, which are the only ones of interest for mathematical physics. As before, we will only consider complex Lie superalgebras. A simple Lie superalgebra is said to be classical if the representation of its even part g0 on its odd part g1 is either irreducible or if it is reducible, it is completely reducible, i.e. can be written as a direct sum of irreducible representations. If gs is a complex classical simple Lie superalgebra, its even part has the form

g = gA gss, (3.49) 0 0 ⊕ 0 A ss where g0 is an Abelian complex Lie algebra and g0 is a semi-simple complex Lie algebra. ss ss Let h0 be a Cartan subalgebra of g0 . Then,

h = gA hss (3.50) s 0 ⊕ 0 is a Cartan subalgebra of gs. The rank r of a classical simple complex Lie algebra is the dimension of its Cartan subalgebra. We can deﬁne the roots analogously to the case of simple Lie algebras. If α(H) is a linear functional on hs, we can ﬁnd at least one element Eα g such that ∈ s [H, Eα] = α(H)Eα H h , (3.51) ∀ ∈ s α and α is called a root of gs, and the set of all elements E that satisfy Eq. (3.51) forms the root subspace g . If Eα g , α is said to be an even root of g , while if Eα g , α is said s,α ∈ 0 s ∈ 1 to be an odd root of gs. The set of all distinct non-zero even roots is denoted by ∆0, and

61 FS2016 Part 3. Advanced topics: Beyond afﬁne Lie algebras

the set of all distinct odd roots is denoted by ∆1. The set of distinct roots of gs contained either in ∆0 or ∆1 or both is denoted by ∆s. hs can be regarded as the subspace of gs corresponding to zero even roots.

The set of classical simple Lie superalgebras can be divided into

1. Basic classical simple Lie superalgebras which posses a non-degenerate bilinear supersymmetric consistent invariant form K0. This set can in turn be divided into

(a) those for which the Killing form K is non-degenerate, so K = K0. (b) those for which the Killing form is identically zero, K = K . 6 0 2. Strange classical simple Lie superalgebras, which do not posses any non-degenerate supersymmetric consistent invariant form.

The following is a complete list of the complex classical simple Lie superalgebras:

1. Basic classical simple Lie superalgebras

(a) with non-degenerate Killing form: i. simple complex Lie algebras ii.• A(r s), r > s 0 | ≥ • B(r s), r > 0, s 1 | ≥ • C(s), s 2 ≥ • D(r s), r 2, s 1, r = s + 1 | ≥ ≥ 6 • F(4) • G(3) (b) with zero Killing form: • A(r r), r 1 | ≥ • D(s + 1 s), s 1 | ≥ • D(2 1; α), α C 0, 1, ∞ | ∈ \{ − } 2. Strange classical simple complex Lie algebras

• P(r), r 2 ≥ • Q(r), r 2. ≥ In the following, we will collect the necessary material in order to deﬁne the Cartan matrices and Dynkin diagrams of basic complex classical simple Lie superalgebras. For semi-simple complex Lie algebras, we have learned that (α, α) > 0 for all non-zero roots α. This is not the case for basic complex classical simple Lie superalgebras:

1. If α ∆ is an even, non-zero root of g , (α, α) = 0, but it need not be real and ∈ 0 s 6 positive. If gs has non-degenerate Killing form, (α, α) R, but can be positive or negative. ∈

2. If α ∆1 is an odd root of gs, it is possible to have (α, α) = 0, even when α is not identically∈ zero.

FS2016 62 Part 3. Advanced topics: Beyond afﬁne Lie algebras

Example: generators and roots of g = A(1 0). A(1 0) is the complex Lie superalgebra s | | sl(2 1; C). Its even part is given by | g = gA gss, (3.52) 0 0 ⊕ 0 A ss where g0 is a one-dimensional Abelian Lie algebra and g0 = A1. The Cartan subalgebra 1 A 1 is two-dimensional with basis elements H1 and C, where C is the generator of g0 and H1 is the generator of hss = h . Let E be the 3 3 matrix with entry (p, q) = 1 and all 0 A1 pq × others zero. The generators of the Cartan subalgebra are given by H1 = E E and 1 11 − 22 C = E11 E22 2E33. E12, E21, E13, E31, E23 and E32 generate the root subspaces. It is easy− to work− out− the commutation relations, we have e.g.

[H1, E ] = 2 E , [H1, E ] = E , [H1, E ] = E , (3.53) 1 12 12 1 13 13 1 23 − 23 [C, E12] = 0, [C, E13] = E13, [C, E23] = E23, (3.54)

[E12, E13] = 0, [E13, E23] = 0, [E12, E23] = E13. (3.55)

A(1 0) has the even roots α associated with E , E and the odd roots α , α associ- | ± 1 12 21 ± 2 ± 3 ated to E23, E32 and E13, E31. We see that α3 = α1 + α2. Taking the basis elements of A(1 0) in the order | C, H1, E , E , E , E , E , E , (3.56) { 1 12 21 13 31 23 32} we can calculate adC, which is a diagonal matrix with elements 0, 0, 0, 0, 1, 1, 1, 1 and 1 { − − } adH1 which is again diagonal with elements 0, 0, 2, 2, 1, 1, 1, 1 . From here, it is easy {1 1 − − − 1} to ﬁnd the Killing forms K(C, C) = 4, K(H1 , H1 ) = 4 and K(C, H1 ) = 0. The roots have the scalar products−

(α1, α1) = 1, (α2, α2) = (α3, α3) = 0, (3.57) (α , α ) = (α , α ) = 1 , (α , α ) = 1 . (3.58) 1 2 2 3 − 2 1 3 2

In the following, we need to deﬁne the concepts of positive, negative and simple roots such that as many results from simple Lie algebras carry over as possible. One main difference however remains. For simple Lie algebras, all choices for the set of positive roots are equivalent, which is not the case for simple Lie superalgebras. ss ss Let h0 be the Cartan subalgebra of the semi-simple Lie algebra g0 , and suppose a ss ss ss choice of positive roots of g0 has been made. Then the subalgebra b0 of g0 that consists of ss ss all elements of h0 together with the root subspaces corresponding to α ∆+ of g0 forms a ss ∈ maximal solvable subalgebra of g0 . If g0 contains a one-dimensional Abelian Lie algebra A g0 , then we deﬁne the subalgebra b0 as b = bss gA, (3.59) 0 0 ∪ 0 A but if g0 = 0 , then { } ss b0 = b0 . (3.60)

In both cases, b0 is a maximal solvable subalgebra of g0 and is known as a Borel subalgebra of g0. Let b be a maximal solvable subalgebra of the Lie superalgebra gs and b0 b. As + ⊂ the Cartan subalgebra is contained in b0, also hs b, and so a subspace gs can be deﬁned such that ⊂ N ⊂ b = h +. (3.61) s ⊕ N

63 FS2016 Part 3. Advanced topics: Beyond afﬁne Lie algebras

We can introduce a further subspace via N − + g = − h . (3.62) s N ⊕ s ⊕ N A root α g is said to be positive if the intersection of the root subspace g with +, ∈ s s,α N g +, is non-trivial and negative if g is non-trivial. This deﬁnition implies s,α ∩ N s,α ∩ N − that if α is positive, α is negative and vice versa. Moreover, every positive even root is − ss ss an extension of a positive root of g0 , and every positive root of g0 extends to a positive even root of gs. In the exceptional case gs = A(1 1), dimgs,α = 2 for every odd root α, and + | gs,α and gs,α − are both non-trivial, implying that for A(1 1), every odd root is both∩positive N and negative.∩ N | A non-zero root α of gs is said to be simple if α is positive, but cannot be expressed as α = β + γ, with β, γ positive roots of gs.

Example: Positive, negative and simple roots of g = A(1 0). We have seen that s | ∆s = α1, α2, α3 . The corresponding root subspaces are all one-dimensional and {± ± ± α } generated by the E± i .

1. One choice of b has basis C, H1, Eα1 , Eα2 , Eα3 , so that + has basis Eα1 , Eα2 , Eα3 . { 1 } N { } With this choice, α , α , α are all positive roots and α , α , α are negative roots. 1 2 3 − 1 − 2 − 3 Since α3 = α1 + α2, the corresponding simple roots are α1 and α2.

2. Another inequivalent choice of b has basis C, H1, Eα1 , E α2 , Eα3 , so that + has { 1 − } N basis Eα1 , E α2 , Eα3 and hence has basis E α1 , Eα2 , E α3 . However, now { − } N − { − − } α3 = α1 + α2 is no longer a sum of positive roots and is hence simple. We rename the positive roots to α10 = α3, α20 = α2, α30 = α1. The simple roots are now α10 and α20 , − 1 both being odd. Note that (α10 , α10 ) = (α20 , α20 ) = 0, (α10 , α20 ) = 2 .

The number of simple roots R is related to the rank r of gs by ( r + 1 if g = A(p, p), p > 0, R = s (3.63) r for any other basic gs.

Moreover, the set of simple roots α1,..., αR is always linearly independent except for A(p, p), p > 0. We can deﬁne the Cartan matrix A of a basic classical simple complex Lie superalgebra g to be a R R matrix with matrix elements A deﬁned in terms of the simple roots as s × jk

2(αj, αk) Ajk = if (αj, αj) = 0, (3.64) (αj, αj) 6

(αj, αk) Ajk = if (αj, αj) = 0, (3.65) (αj, αj0 ) where α is another simple root such that (α , α ) = 0. In the ﬁrst case, A = 2 and the j0 j j0 6 jj only possible values of Ajk, j = k are 0, -1, -2 and -3. In the second case, Ajj = 0 and , which is clearly very different6 from the allowed values of a Cartan matrix of an Ajj0 = 1 ordinary simple Lie algebra. A is an r r matrix except for A(p, p), p > 0. For D(2 1; α) with α / R, at least one × | ∈ off-diagonal element of A is not real.

FS2016 64 Part 3. Advanced topics: Beyond afﬁne Lie algebras

Example: Cartan matrices of of g = A(1 0). With choice (1) for the simple roots, we s | had (α , α ) = 0, but (α , α ) = 0, so we can take α = α . Then, 2 2 2 1 6 20 1 2 1 A = − . (3.66) 1 0 With choice (2) for the simple roots, taking and , we ﬁnd α20 = α1 α10 = α2 0 1 A = . (3.67) 1 0 F

We want to again associate to each Cartan matrix a generalized Dynkin diagram. They are constructed according to the following rules:

1. assign to each simple root αj a vertex j which is drawn as

• if αj is even. This is called a white node.

• if αj is odd and (αj, αj) = 0. This is called a grey node. • if α is odd and (α , α ) = 0. This is called a black node. j j j 6

2. draw ljk lines from the vertex j to the vertex k, where l = max A , A . (3.68) jk {| jk| | kj|} 3. add an arrow pointing from the j vertex to the k vertex if A > 1 (except for | kj| D(2 1; α). |

Example: Generalized Dynkin diagrams of gs = A(1 0). The generalized Dynkin diagram corresponding to the choice (1) is |

1 2 (3.69) The generalized Dynkin diagram corresponding to the choice (2) is

1 2 (3.70)

While each simple complex Lie algebra has a unique Dynkin diagram, we see that this uniqueness is lost in the superalgebra case. Due to the anomalous cases D(2 1; α) and | D(2 s), s > 0, the above prescription cannot be reversed to determine the Cartan matrix from| the Dynkin diagram. Therefore, the Dynkin diagrams play a less important role for superalgebras.

For each basic classical simple complex Lie superalgebra gs with non-trivial odd part, there is a distinguished choice of simple roots, which consists of one odd root, all other R 1 simple roots being even. The corresponding Dynkin diagram therefore has R 1 white− vertices and one grey or black vertex. In Table 3.2, all the Dynkin diagrams of− the basic classical simple complex Lie superalgebras corresponding to the distinguished choice of the Borel subalgebra are depicted.

65 FS2016 Part 3. Advanced topics: Beyond afﬁne Lie algebras

1 2 r r + 1 r + s + 1 A(r s) r s > 0 | ≥

1 2 r r + 1 2r + 1 A(r r) r 1 | ≥

1 2 r r + 1 r 1 r B(r s), r > 0, s > 0 − |

1 2 s 1 s B(s), s > 0 −

C(s), s > 0 1 2 s 1 s − r + s D(r s), r > 2, s > 0 | r + s 2 1 2 s 1 s − − r + s 1 − s + 2

D(2 s), s > 0 s | 1 2 s + 1 3

D(2 1; α), α = 1, 0, ∞ 1 | 6 − 2

F 4 1 2 3 4

G 2 1 2 3

Table 3.2: Basic classical simple complex Lie superalgebras corresponding to the distinguished choice of the Borel subalgebra

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3.3 Quantum groups

In this last section of the course, we touch on a subject which requires the highest level of abstraction encountered so far, namely quantum groups. They act as a generalized symmetry and arise naturally in several independent contexts in theoretical physics, one of which is the study of integrable models. The concept of quantum group is mathematically speaking equivalent to the one of the Hopf algebra, which is an associative algebra with a host of extra structure. We will concentrate on a particular class of Hopf algebras, namely Uq(g), a deformation of the enveloping algebra U(g) of a complex simple Lie algebra g. Uq(g) depends in a natural way on the algebra g alone, so the theory of Uq(g) can be viewed as a self-contained outgrowth of Lie algebra theory. It should be pointed out, however, that the theory of general quantum groups is much richer and encompasses new phenomena that have no analogy in Lie algebra theory.

The fundamental new concept to be introduced is the one of a Hopf algebra. Let a be a vector space and F the base field of the vector space (in the following, we take again F = C). A Hopf algebra a is a vector space endowed with five operations : a a a (multiplication) (3.71) M × → η : F a (unit map) (3.72) → ∆ : a a a (co-multiplication) (3.73) → × e : a F (co-unit map) (3.74) → γ : a a (antipode), (3.75) → which have the following properties: (id ) = ( id) (associativity) (3.76) M ◦ × M M ◦ M × (id η) = id = (η id) (existence of unit) (3.77) M ◦ × M ◦ × (id ∆) ∆ = (∆ id) ∆ (co-associativity) (3.78) × ◦ × ◦ (e id) ∆ = id = (id e) ∆ (existence of co-unit) (3.79) × ◦ × ◦ (id γ) ∆ = η e = (γ id) ∆ (3.80) M ◦ × ◦ ◦ M ◦ × ◦ ∆ = ( ) (∆ ∆) (connecting axiom). (3.81) ◦ M M × M ◦ ◦ At first sight, this structure might seem quite complicated, but we will see that it arises quite naturally. Vector spaces with a multiplication satisfying Eq. (3.76) are called associative algebras and are very common. Many of them have a unit (unital algebras). By formalizing the notion of unmultiplying objects, one gets the operations of co- multiplication and co-unit. The co-multiplication of an element X a is the sum of all ∈ those things in a a which could give X when combines according to an underlying group structure. ⊗ Vector spaces endowed with a co-multiplication are called co-algebras. the theory of co-algebras is dual to the one of algebras, hence it is not necessary to consider them in their own right. The requirement of co-associativity is also natural. Having co-multiplication as a dual operation to multiplication, it makes sense to require also the existence of a co-unit as the dual operation to the unit. An algebra a that possesses the four operations , η, ∆ M and e is called a bi-algebra. Finally, we need a way of connecting the operations of multiplication and co-multiplication non-trivially. We need to construct a map as the composition of multiplication and co- multiplication, which leads to the concept of the antipode, and hence to Hopf algebras. The

67 FS2016 Part 3. Advanced topics: Beyond afﬁne Lie algebras

antipode is a weaker structure than the inverse, it provides a nonlocal linearized inverse. It means that now not individual elements, but certain linear combinations are invertible. We can express the deﬁning properties Eq. (3.76)–Eq. (3.81) more explicitly in terms of the elements X a. If we use for the multiplication , ∈ M : X Y (X Y) X Y, (3.82) M ⊗ 7→ M ⊗ ≡ Eq. (3.76) reads now

X (Y Z) = (X Y) Z X, Y, Z a. (3.83) ∀ ∈ Similarly, the existence of a unit means that there is an element E a (the unit element) such that ∈ E X = X = X E X a. (3.84) ∀ ∈ The map η is then given by η : ξ ξE ξ F. (3.85) 7→ ∀ ∈ Another way of expressing the properties Eq. (3.76)–Eq. (3.81) in a more explicit way is via commutative diagrams. A diagram is said to be commutative iff the composite maps which are obtained by following the arrows are independent of the path used to link any two given spaces in the diagram. The following diagram shows the associativity property Eq. (3.76).

a a id M × × M a a a a (3.86) × × M id M × a a × The existence of the unit (Eq. (3.77)) is shown below, with the map s : a F a the scalar multiplication. × → a F s a F a × s × id M M id (3.87) × × a a M a a a × M × Co-associativity (Eq. (3.78)) is shown below:

a a ×

id ∆ ∆ × a a a a (3.88) × × ∆ id ∆ × a a × Finally, the existence of the co-unit (Eq. (3.79)) is expressed as follows, with the map i : a a F the inclusion X X 1, where 1 is the multiplicative unit of F. → × 7→ ⊗ a F a i F a × i × id e e id (3.89) × × a a a ∆ a a × ∆ ×

FS2016 68 Part 3. Advanced topics: Beyond afﬁne Lie algebras

We see that the third and fourth diagrams are obtained from the ﬁrst and second by reversing the arrows, in other words, the rules for ∆ are just the same as the rules for multiplication, with the arrows reversed. When a basis Ja of a is ﬁxed, then multiplication, co-multiplication and antipode can be expressed in terms{ } of structure constants, just like the bracket relations of a Lie algebra:

a b ab c J J = ∑ µc J , (3.90) c c c a b ∆(J ) = ∑ νab J J , (3.91) a,b ⊗ c c a γ(J ) = ∑ τa J . (3.92) a The associativity property Eq. (3.76) is then expressed as

bc ad ab dc µd µ f = µd µ f . (3.93) Similarly, the structural properties Eq. (3.78) and Eq. (3.80) read

a b a b νcdνe f = νecνf d, (3.94) a c bd a d bc νbcτd µe = νdcτb µe . (3.95)

Example: Universal enveloping algebra U(g) of a complex Lie algebra g. The universal enveloping algebra U(g) of a Lie algebra g consists of all ﬁnite formal power series in the elements of g. U(g) is an associative algebra with the product given by termwise formal multiplication. As a vector space, it is generated by all monomials in the generators of g, identifying however all monomials which become equal to each other upon use of the bracket relations of g. U(g) becomes a Hopf algebra by taking as the usual formal multiplication on U(g), and deﬁning the unit element by M

η(ξ) = ξ1 ξ C. (3.96) ∀ ∈ Co-multiplication, co-unit and antipode are deﬁned by

∆(X) = X 1 + 1 X, (3.97) ⊗ ⊗ e(X) = 0, (3.98) γ(X) = X (3.99) − and

∆(1) = 1 1, (3.100) ⊗ e(1) = 1, (3.101) γ(1) = 1. (3.102) − Note, that in this case, γ γ = id. In particular, for any X g, ξ C, the formal element ◦ ∈ ∈ X˜ = eξX U(g) obeys ∈ ∆(eξX) = eξX eξX, (3.103) ⊗ ξX ξX γ( ) = e− . (3.104) As a consequence, this element satisﬁes

X˜ γ(X˜ ) = 1 = γ(X˜ ) X˜ , (3.105)

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which justiﬁes the description of the antipode as the analogue of an inverse.

Some general properties of Hopf algebras are the following:

• For a given multiplication and co-multiplication, the co-unit is unique.

• If (a, , η, ∆, e, γ) is a Hopf algebra, then the dual vector space a inherits a Hopf M ∗ algebra structure by interchanging , η with ∆, e. M • The co-multiplication and co-unit are homomorphisms of a, i.e. preserve the multiplication. For the co-unit, this means

e(X Y) = e(X)e(Y), X, Y a, (3.106) ∀ ∈ and for the co-multiplication,

∆(X Y) = ∆(X) ∆(Y), X Y a a. (3.107) ∀ ⊗ ∈ ×

• The antipode is an anti-homomorphism, i.e.

γ(X Y) = γ(Y)γ(X), X, Y a. (3.108) ∀ ∈

• The antipode is an anti-cohomomorphism, i.e.

(γ γ) ∆ = π ∆ γ, (3.109) × ◦ ◦ ◦ where π is the permutation map

π : a a a a (3.110) × → × X Y Y X. (3.111) ⊗ → ⊗

• The map ∆ := π ∆ is also a co-associative multiplication. 0 ◦ An algebra a is said to be commutative iff the multiplication does not depend on the order of the factors, π = . (3.112) M ◦ M Analogously, a Hopf algebra is called co-commutative iff the co-multiplication satisﬁes

π ∆ = ∆, (3.113) ◦

or in other words, iff ∆0 coincides with ∆. An example of a co-commutative Hopf algebra is given by the universal enveloping algebras U(g) we have discussed above. For any commutative or co-commutative Hopf algebra, the antipode obeys γ γ = id. ◦ Before we can study the class of quantum universal enveloping algebras Uq(g), we must discuss one last property of Hopf algebras, namely quasitriangularity.A quasitriangular Hopf algebra is a Hopf algebra for which the co-multiplications ∆ and ∆0 are related by conjugation, i.e. 1 ∆0(X) = R ∆(X) R− X a (3.114) ∀ ∈

FS2016 70 Part 3. Advanced topics: Beyond affine Lie algebras for some element R a a which is invertible and satisfies ∈ × (id ∆)(R) = R R , (3.115) × 13 12 (∆ id)(R) = R R , (3.116) × 13 23 1 (γ id)(R) = R− . (3.117) × Here, the inverse R 1 of R a a is by definition that element of a a which satisfies − ∈ × × 1 1 R− R = E E = R R− . (3.118) ⊗ Generally, R has the structure (l) (l) R = ∑ R1 R2 . (3.119) l ⊗ In the above definition, R is meant as the identity in the second factor of a a a and as 13 × × R on the first and third factors, analogously for R12, R23. A quasitriangular Hopf algebra is called triangular iff R R = E E. (3.120) 12 21 ⊗ The universal enveloping algebra U(g) of any Lie algebra g is quasitriangular. Generally, a quasitriangular Hopf algebra is neither commutative nor co-commutative; however, the non-commutativity is under control, see Eq. (3.114). As a consequence, many properties of co-commutative Hopf algebras generalize to generic quasitriangular Hopf algebras. An immediate consequence of Eq. (3.114) is that

R (∆ id)(X Y) = (∆0 id)(X Y) R X Y a a. (3.121) 12 × ⊗ × ⊗ 12 ∀ ⊗ ∈ × Taking X Y = R and using Eq. (3.115) and (3.116), this yields ⊗ R R R = R R R . (3.122) 12 13 23 23 13 12 This is the so-called Yang–Baxter equation which plays a fundamental role in the theory of completely integrable systems. In this context, R is called the universal R-matrix. The quantum universal enveloping algebra Uq(g) is the algebra of power series in the 3r + 1 generators ei, f i, hi i = 1, . . . , r 1 (3.123) { | } ∪ { } modulo the relations

[hi, hj] = 0, (3.124) i j j [h , e ] = Ajie , (3.125) [hi, f j] = A f j, (3.126) − ji [ei, f j] = δij hi , (3.127) b c

1 Aji − p 1 Aji i p j i 1 A p ∑ ( 1) − (e ) (e )(e ) − ji− = 0 i = j, (3.128) p=0 − p i 6

1 Aji − p 1 Aji i p j i 1 A p ∑ ( 1) − ( f ) ( f )( f ) − ji− = 0 i = j, (3.129) p=0 − p i 6 1 X = X = X 1 X U (g), (3.130) ∀ ∈ q

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where A are the elements of the Cartan matrix of g. Here, [X, Y] X Y Y X with ji ≡ − (X Y) = X Y the formal product in U(g). The q-number symbol is deﬁned by M ⊗ b c qX/2 q X/2 X X := − − , (3.131) b c ≡ b cq q1/2 q 1/2 − − together with

X X with q q(αi,αi)/(θ,θ), (3.132) b ci ≡ b cqi i ≡ n j n k n ! n !:= m , := b c . (3.133) b c ∏ b c m m ! n m ! m=1 b c b − c The exponential functions of generators appearing in hi are defined via the corresponding power series, i.e. b c ∞ ξn eξh = ∑ hn (3.134) n=0 n! with hn h n defined inductively, i.e. h n = h h (n 1), so that in particular ≡ − ξh ξh e e− = 1. (3.135) hi/2 Note that due to the appearance of q± , we are forced to consider infinite power series in the hi. In contrast, it is consistent to restrict to only finite power series in the ei, f i. In the limit q 1, the relations Eq. (3.124) to (3.127) reduce to the Lie brackets of g and the → relations Eq. (3.128), (3.129) to the Serre relations of g.

Example: Serre relations for A = 1. In this case, the relation Eq. (3.128) is given by ji − i i j 1/2 1/2 i j i j i i e e e (q + q− )e e e + e e e = 0, (3.136) − i 1/2 1/2 similarly for f . In the limit q 1, we have (q + q− ) 2 = 2, so that Eq. (3.136) reduces to → ≡ b c 0 = eieiej 2 eiejei + ejeiei = [ei, [ei, ej]] = (ad(ei))2ej. (3.137) − F

We can deﬁne the quantum version of the operator ad(X) as

α β α β (α,β)/4 α β (α,β)/4 β α Ad(E )E [E , E ] := q− E E q E E . (3.138) ≡ q −

Example: Uq(A1). The relations Eq. (3.128)-(3.129) become more transparent in the simplest example Uq(A1), where there are no Serre relations. The deﬁning relations of Uq(A1) are

[h, e] = 2 e, (3.139) [h, f ] = 2 f , (3.140) − [e, f ] = h . (3.141) b c

The enveloping algebra U(A1) is by deﬁnition associative and its unit 1 is given by the trivial power series 1. These properties are inherited by Uq(A1). Uq(A1) is moreover

FS2016 72 Part 3. Advanced topics: Beyond afﬁne Lie algebras endowed with the structure of a quasitriangular Hopf algebra. The co-multiplication acts on the generators h, e, f and on 1 as

∆(h) = h 1 + 1 h, (3.142) ⊗ ⊗ h/4 h/4 ∆(e) = e q + q− e, (3.143) ⊗ ⊗ h/4 h/4 ∆( f ) = f q + q− f , (3.144) ⊗ ⊗ ∆(1) = 1 1, (3.145) ⊗ and the antipode is

γ(h) = h, (3.146) − γ(e) = q1/2e, (3.147) − γ( f ) = q1/2 f , (3.148) − γ(1) = 1. (3.149)

Finally, the co-unit is given by

e(h) = e(e) = e( f ) = 0, (3.150) e(1) = 1. (3.151)

We need the identity

[qξh, e] = (q2ξ 1)e qξh, [qξh, f ] = (q2ξ 1) f qξh (3.152) − − to verify the deﬁning property of γ. The antipode γ associated to ∆ = π ∆ is obtained 0 0 ◦ from γ by τ τ 1 (see Eq. (3.92)). The quantum group U (A ) deﬁned by ∆ and γ is → − q 1 0 0 0 related to Uq(A1) by U (A ) = U 1 (A ). (3.153) q 1 0 q− 1 F

The Hopf algebra structure of Uq(A1) generalizes in a natural way to Uq(g) with arbitrary simple or even afﬁne g. The deﬁning formulae for the co-multiplication are

∆(hi) = hi 1 + 1 hi, (3.154) ⊗ ⊗ i i hi/4 hi/4 i ∆(e ) = e q + q− e , (3.155) ⊗ ⊗ i i hi/4 hi/4 i ∆( f ) = f q + q− f , (3.156) ⊗ ⊗ ∆(1) = 1 1. (3.157) ⊗ For the co-unit,

e(hi) = e(ei) = e( f i) = 0, (3.158) e(1) = 1, (3.159) and for the antipode,

γ(hi) = hi, (3.160) − i h i h γ(e ) = q ρ e q− ρ , (3.161) − i h i h γ( f ) = q ρ f q− ρ , (3.162) − γ(1) = 1, (3.163)

73 FS2016 Part 3. Advanced topics: Beyond afﬁne Lie algebras

where 1 1 r r h = h α hi = ρ˜ hi = (ρ˜ h) (3.164) ρ ∑ α∨ ∑ ∑ i ∑ i , , (θ, θ) α>0 ≡ (θ, θ) α>0 i=1 i=1 with 2 , the Weyl vector of . ρ˜ := (θ,θ) ρ ρ g It is straightforward to check that the maps deﬁned above satisfy the deﬁning properties of a Hopf algebra. To check the co-associativity of ∆, we use ∆(qξh) = qξh qξh. (3.165) ⊗ Above, we have only presented the Hopf algebra structure for the Chevalley generators. To identify it also for the rest of the generators, we have to use the Serre relations together with the homomorphism property of the co-multiplication, etc.

(α +α ) Example: Hopf algebra action on E± 1 2 in Uq(A2). Using Eq. (3.138), we ﬁnd

(α +α ) 1/4 α α 1/4 α α E± 1 2 = q E± 1 E± 2 q− E± 2 E± 1 . (3.166) − Using the homomorphism property of ∆, its action on the Chevalley generators and the commutation relations of Uq(A2), we ﬁnd

(α +α ) (α +α ) (h1+h2)/4 (h1+h2)/4 (α +α ) ∆(E 1 2 ) =E 1 2 q + q− E 1 2 (3.167) ⊗ ⊗ 1/2 1/2 h1/4 α h2/4 α + (q + q− )q− E 2 q E 1 , ⊗ (α +α ) (α +α ) (h1+h2)/4 (h1+h2)/4 (α +α ) ∆(E− 1 2 ) =E− 1 2 q + q− E− 1 2 (3.168) ⊗ ⊗ 1/2 1/2 h2/4 α h1/4 α + (q + q− )q− E− 1 q E− 2 . ⊗ Similarly, we ﬁnd for the antipode

(α +α ) 1 1/4 α α 1/4 α α γ(E 1 2 ) = q± q− E± 1 E± 2 q E± 2 E± 1 . (3.169) − F

Literature. The Virasoro algebra is brieﬂy introduced in [FS97], and at much more length but from a completely different starting point (i.e. conformal ﬁeld theory) in [DMS97]. The section on Lie superalgebras is summarizing parts of [Cor89], which is very extensive. The part on quantum groups is taken mostly from [Fuc92], supplemented by [Maj95].

References

[FS97] J. Fuchs and C. Schweigert. Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists. Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press, 1997. ISBN: 978-0521541190. [DMS97] P. Di Francesco, P. Mathieu, and D. Sénéchal. Conformal field theory. Graduate texts in contemporary physics. New York: Springer, 1997. ISBN: 0-387-94785-X. URL: http: //opac.inria.fr/record=b1119694. [Cor89] J. Cornwell. Group Theory in Physics, Volume III. Supersymmetries and Infinite-Dimensional Algebras. Techniques of Physics. Academic Press, 1989. ISBN: 0-12-189805-9. [Fuc92] J. Fuchs. Affine Lie Algebras and Quantum Groups: An Introduction, with Applications in Conformal Field Theory. 1st. Cambridge Monographs on Mathematical Physics. Cam- bridge University Press, 1992. ISBN: 0521415934. [Maj95] S. Majid. Foundations of Quantum Group Theory. Cambridge: Cambridge University Press, 1995. ISBN: 0-521-46032-8.

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Epilogue: Lie algebras in integrable spin chains

Despite having treated a mathematical subject, this course is intended for theoretical physicists. As this course did not contain any physics examples, let me close by giving three examples of integrable spin chains which have a simple Lie algebra, a quantum group and a Lie superalgebra as symmetry algebras. Of course this is just one context in which these algebras arise, but it is a particularly simple one. We will see that small modiﬁcations of a simple physical problem, such as turning on an external magnetic ﬁeld or allowing lattice sites to be empty, changes the structure of the symmetry algebra of the system from the one of a simple Lie algebra to being a quantum group or a Lie superalgebra. All these examples are completely solvable and a beautiful subject in themselves, however we won’t touch upon how to actually solve them here.

The XXX1/2 or Heisenberg spin chain. Let us consider a closed linear chain of L identical atoms with only next-neighbor interactions. Each atom has one electron in an outer shell (all other shells being complete). These electrons can either be in the state of spin up ( ) or down ( ). At ﬁrst order, the Coulomb- and magnetic interactions result in the exchange↑ interaction↓ in which the states of neighboring spins are interchanged:

. (3.170) ↑↓ ↔ ↓↑ In a given spin conﬁguration of a spin chain, interactions can happen at all the anti-parallel pairs. Take for example the conﬁguration

. (3.171) ↑↑↓↑↓↓↓↑↑↓↓ It contains five anti-parallel pairs on which the exchange interaction can act, giving rise to five new configurations. The Hamiltonian of the XXX1/2 spin chain is given by

L = J ∑ Πn,n+1, (3.172) H − n=1

1 where J is the exchange integral and Πn,n+1 is the permutation operator of states at positions n, n + 1. The spin operator at position n on the spin chain is given by ~ x y z 1 Sn = (Sn, Sn, Sn) = 2~σn, (3.173) i where σn are the Pauli matrices for spin 1/2: 0 1 0 i 1 0 σ1 = , σ2 = − , σ3 = . (3.174) 1 0 i 0 0 1 − 1 J > 0: ferromagnet, spins tend to align, J < 0: anti-ferromagnet, spins tend to be anti-parallel. − −

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For the closed chain, the sites n and n + L are identiﬁed:

~SL+1 = ~S1. (3.175)

In terms of the spin operators, the permutation operator is given by

Π = 1 (1 +~σ ~σ ). (3.176) n,n+1 − 2 n n+1 The Hamiltonian (3.172) now becomes

L = J ~S ~S H − ∑ n n+1 n=1 (3.177) L 1 + + z z = J ∑ 2 Sn Sn−+1 + Sn−Sn+1 + SnSn+1, − n=1

x y where S± = Sn iSn are the spin ﬂip operators. The term in parentheses corresponds to the exchange interaction± which exchanges neighboring spin states. The spin ﬂip operators act as follows on the spins:

S+ ...... = 0, S+ ...... = ...... , k | ↑ i k | ↓ i | ↑ i S− ...... = ...... , S− ...... = 0, (3.178) k | ↑ i | ↓ i k | ↓ i Sz ...... = 1 ...... , Sz ...... = 1 ...... k| ↑ i 2 | ↑ i k| ↓ i − 2 | ↓ i The spin operators have the commutation relations

z + z [S , S±] = S±δ , [S , S−] = 2S δ , (3.179) n m ± n nm n m n nm

which we recognize as those of A1. The Hamiltonian moreover commutes with the generators of su(2), z [H, S±] = [H, S ] = 0, (3.180)

thus we say that su(2) is the symmetry algebra of the XXX1/2 spin chain and each lattice site carries a spin 1/2 representation of su(2).

The XXZ1/2 or anisotropic spin chain. In the anisotropic case, a magnetic ﬁeld is turned on in the z-direction, resulting in the Hamiltonian

L x x y y z z 1 ∆ = J ∑ SnSn+1 + SnSn+1 + ∆(SnSn+1 4 ). (3.181) H − n=1 −

The anisotropy is captured by the parameter

q + q 1 ∆ = − . (3.182) 2 ∆ = 1 is the isotropic case we have treated so far. The XXZ spin chain admits the quantum group Uq(su(2)) as symmetry algebra. Uq(su(2)) + Sz is generated by S , S− and q± under the relations

2Sz 2Sz Sz Sz 1 + q q− q S±q = q± S±, [S , S−] = − . (3.183) q q 1 − −

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These relations reduce to the ones of A1 for q 1. For the case of spin 1/2, we ﬁnd the following representations for the operators: →

z 3 3 qS = qσ /2 qσ /2, (3.184) | ⊗ ·{z · · ⊗ } L L L σ3/2 σ3/2 σn± σ3/2 σ3/2 S± = ∑ Sn± = ∑ q q q− q− , (3.185) n=1 n=1 | ⊗ ·{z · · ⊗ } ⊗ 2 ⊗ ⊗ · · · ⊗ n 1 − where we have the Pauli matrices 0 1 0 0 1 0 σ+ = , σ = , σ3 = . (3.186) 0 0 − 1 0 0 1 − and 3 q 0 qσ = . (3.187) 0 1/q

The tJ model. The tJ model describes a system of electrons on a lattice with a Hamiltonian that describes nearest–neighbor hopping (with coupling t) and spin interactions (with coupling J). Consider a one-dimensional lattice of length L with periodic boundary conditions. Each site can be either free ( ) or occupied by a spin up ( ) or down ( ) ◦ ↑ ↓ electron. Excluding double occupancy, the Hilbert space at each point k is:

(1 2) Hk = C | . (3.188)

† It is convenient to introduce anticommuting creation–annihilation pairs ck,s, ck,s, s = , at each site, acting as {↑ ↓}

s = c† , for s = , , (3.189) | ik k,s|◦ik {↑ ↓} where is the vacuum, annihilated by c . Let n = c† c be the number of s electrons |◦ik k,s k,s k,s k,s at position k and nk = nk, + nk, . We can further introduce su(2) spin operators at each site: ↑ ↓

† + † z 1 Sk− = ck, ck, , Sk = ck, ck, , Sk = 2 nk, nk, . (3.190) ↑ ↓ ↓ ↑ ↑ − ↓ With these ingredients, we can write down the Hamiltonian " # L 1 − † ~ ~ 1 1 = ∑ t ∑ ck,sck+1,s + h.c. + J Sk Sk+1 4 nknk+1 + 2 nk 2 , H k=1 − P s= , P · − − ↑ ↓ (3.191) where projects out double occupancy. P For J = 2t = 2, the Hamiltonian is invariant under the action of the Lie superalgebra sl(1 2). | z The even part g0 = gl(1) sl(2) of sl(1 2) is generated by the operators S±, S , Z with commutation relations ⊕ |

z + z z [S , S±] = S± , [S , S−] = 2S , [Z, S±] = 0 , [Z, S ] = 0 . (3.192) ± There are two additional fermionic multiplets Q , s = , which transform with respect s± {↑ ↓} to g0 as

z 1 1 1 [S , Qs±] = 2 Qs , [S±, Qs±] = 0 , [Z, Q±] = 2 Q± , [Z, Q±] = 2 Q± . (3.193) ± ↓ ↓ ↑ − ↑

77 FS2016 Part 3. Advanced topics: Beyond afﬁne Lie algebras

The fermionic generators satisfy the following anticommutation relations:

z Qs±, Qs∓ = 0 , Q±, Q± = S± , Q±, Q∓ = Z S . (3.194) { } { ↑ ↓ } { ↑ ↓ } ±

At each point k of the lattice, also the generators Q±, Z can be represented in terms of creation–annihilation operators as

+ † Qk−, = 1 nk, ck, , Qk, = 1 nk, ck, , Qk−, = 1 nk, ck, , (3.195) ↑ − ↓ ↑ ↑ − ↓ ↑ ↓ − ↑ ↓ + † 1 Qk, = 1 nk, ck, , Zk = 1 2 nk . (3.196) ↓ − ↑ ↓ − The supersymmetric Hamiltonian can be expressed in terms of these generators as

L h + + i = ∑ ∑ Qk,sQk−+1,s + Qk+1,sQk−,s + H − k=1 s= , ↑ ↓ L + + z z 1 1 + ∑ Sk Sk−+1 + Sk−Sk+1 + 2SkSk+1 2ZkZk+1 k k+1 . (3.197) k=1 − −

Literature. The XXX1/2 spin chain was introduced by Hans Bethe [Bet31]. The quantum group structure of the XXZ chain is discussed in [PS90]. The tJ model and its supergroup structure are discussed in [EK92].

References

[Bet31] H. Bethe. Zur Theorie der Metalle. Zeitschr. f. Phys. 71 (1931), p. 205. [PS90] V. Pasquier and H. Saleur. Common Structures Between Finite Systems and Conformal Field Theories Through Quantum Groups. Nucl.Phys. B330 (1990), p. 523. [EK92] F. H. L. Essler and V. E. Korepin. A New solution of the supersymmetric T-J model by means of the quantum inverse scattering method (1992).

Acknowledgments

I would like to thank all those students who have made these lecture notes better by pointing out typos. Special thanks go to Manuel Meyer who has brought me a list of typos every week.

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