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Affine Lie Algebras Groups and Algebras for Theoretical Physics Masters course in theoretical physics at The University of Bern Spring Term 2016 R SUSANNE REFFERT Contents 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 1 FS2016 Part 1 Complex semi-simple Lie Algebras Symmetries, and with them, groups and algebras are of paramount importance in theo- retical 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) 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) 2 2 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 f g 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 Lief algebras.g In the following,2 f g we 8 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 Ea of g are chosen such that they satisfy [Hi, Ea] = aiEa. (1.6) The vector a = (a1,..., ar) is called a root2. The Ea 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 a naturally maps an element Hi h to the number ai: 2 a(Hi) = ai. (1.7) The roots are therefore elements of the dual of the Cartan subalgebra, a h∗. (1.8) 2 a † a With (E ) = E− , we see from hermitian conjugation of Eq. (1.6) that whenever a is a root, so is a. We use the notation − D = set of all roots , (1.9) f g also called the root system. The root components ai 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 ai are roots of the characteristic equation for Hi. 3 FS2016 Part 1. Complex semi-simple Lie Algebras the vector space on which the generators act. A matrix representation of the adjoint representation in the basis Ja is given by f g (Ja) = i f . (1.10) bc − abc In the adjoint representation, we have Ea Ea a , (1.11) 7! j i ≡ j i Hi Hi , (1.12) 7! j 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)Ea = aiEa Hi a = ai a . (1.14) 7! j i j i The one-to-one correspondence between the a and the Ea reflects the fact that the roots j i are non-degenerate. In the adjoint representation, the zero eigenvalue has degeneracy r (associated to the Hi ). j 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, Ea . From the Jacobi identity, we find f g [Hi, [Ea, Eb]] = (ai + bi)[Ea, Eb]. (1.16) Therefore, 8 [Ea, Eb] ∝ Ea+b if a + b D, <> 2 [Ea, Eb] = 0 if a + b / D, (1.17) > 2 :[Ea, Eb] is a linear comb. of Hi if a = b. − We will use in the following the expressions r r a H = ∑ ai Hi, a 2 = ∑ aiai. (1.18) · i=1 j j i=1 The full set of commutation relations of g in the Cartan–Weyl basis is given by [Hi, Hj] = 0, (1.19) [Hi, Ea] = aiEa (1.20) 8 N Ea+b if a + b D <> a,b 2 a b 2 if [E , E ] = a2 a H a = b (1.21) > j j · − :0 otherwise, where Na,b = 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. FS2016 4 Part 1. Complex semi-simple Lie Algebras 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) ◦ 2 The Killing form obeys X, Y g 8 2 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) 8 ) This is an alternative way of defining semi-simplicity.
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