INTRODUCTION to QUANTUM GROUPS 1. Basic Notions of Lie Algebras Work Over the Field C of Complex Numbers. 1.1. Lie Algebras

INTRODUCTION TO QUANTUM GROUPS

1. Basic notions of Lie algebras Work over the ﬁeld C of complex numbers. 1.1. Lie algebras. Deﬁnition 1.1. A Lie algebra g is a vector space endowed with a bilinear map [ , ]: g × g −→ g, called the Lie bracket, which has the following properties: for all x, y, z ∈ g, [x, y] = −[y, x], (skew symmetry)(1.1) [[x, y], z] + [[y, z], x] + [[z, x], y] = 0. (Jacobi idenity)(1.2) Call [x, y] the commutator of x, y. A Lie algebra g is called commutative or abelian if all commutators are zero, that is, [g, g] = {0}. Two Lie algebras g and g0 are called isomorphic if there is a vector space iso- morphism ϕ : g −→ g0 such that ϕ([x, y]) = [ϕ(x), ϕ(y)] for all x, y ∈ g. A Lie subalgebra k of a Lie algebra g is a subspace such that [k, k] ⊆ k.

Example 1.2. The general linear Lie algebra gln(C) is the space of n × n matrices (over C) with the Lie bracket deﬁned by

[X,Y ] = XY − YX, ∀X,Y ∈ gln(C). Skew symmetry of the Lie bracket is clear. The Jacobi identity follows from the associativity of matrix multiplication. The general linear Lie algebra may be formulated more abstractly. Let V be any vector space and denote by EndC(V ) the vector space of all endomorphisms, i..e, the space of linear maps V → V . Then the general linear Lie algebra of V is gl(V ) = EndC(V ) with the Lie bracket deﬁned by [X,Y ] = XY − YX, ∀X,Y ∈ gl(V ). Exercise 1.3. Show that any associative algebra A is a Lie algebra with the Lie bracket deﬁned by [ , ]: A × A −→ A, [a, b] = ab − ba ∀a, b ∈ A.

Example 1.4. Let sln(C) ⊆ gln(C) be the subspace of traceless matrices. Then sln(C) is a subalgebra of gln(C), which is called the special linear Lie algebra. In particular, the Lie algebra sl2(C) has a basis 0 1 0 0 1 0 E = ,F = ,H = , 0 0 1 0 0 −1 with the following commutation relations (1.3) [H,E] = 2E, [H,F ] = −2F, [E,F ] = H. 1 2 QUANTUM GROUPS

Example 1.5. Let b be the subspace of upper triangular matrices, and h be the subspace of diagonal matrices, in gln(C). Then h ⊆ b ⊆ gln(C) are subalgebras.

Exercise 1.6. Prove the commutation relations (??) of sl2(C).

Exercise 1.7. Let eij (i, j = 1, 2, . . . , n) be the n × n matrices such that the (i, j)- entry of eij is 1 and all other entries are 0. Call these matrices the n × n matrix units. Prove the following results.

(1) The matrix units eij (i, j = 1, 2, . . . , n) form a basis of gln. (2) The matrix units satisfy eijek` = δjkei` for all i, j, k, `. (3) The general linear Lie algebras gln has the following commutation relations eq:gl (1.4) [eij, ek`] = δjkei` − δ`iekj, ∀i, j, k, `.

Exercise 1.8. The subspace of gln consisting of matrices with vanishing n-th row and n-th column is a Lie subalgebra, which is isomorphic to gln−1. Thus we have the chain of subalgebras

gl1 ( gl2 ( gl3 ( ··· ( gln−1 ( gln. Example 1.9. Let V be a vector space equipped with a non-degenerate symmetric bilinear form ω : V × V −→ C. Consider the subspace so(V ; ω) := {x ∈ gl(V ) | ω(x(v), w) + ω(v, x(w)) = 0, ∀v, w ∈ V }. For any x, y ∈ so(V, ω), we have ([x, y](v), w) =(xy(v) − yx(v), w) = −(y(v), x(w)) + (x(v), y(w)) =(v, yx(w)) − (v, xy(w)) = −(v, [x, y](w)), ∀v, w ∈ V. Hence this forms a Lie algebra, called the orthogonal Lie algebra. Over C the orthogonal Lie algebras corresponding to different choices of non- degenerate symmetric bilinear forms are isomorphic. Example 1.10. Let V be a vector space equipped with a non-degenerate skew symmetric bilinear form ω : V × V −→ C. Then the subspace sp(V ; ω) := {x ∈ gl(V ) | ω(x(v), w) + ω(v, x(w)) = 0, ∀v, w ∈ V } forms a Lie algebra, called the symplectic Lie algebra. Note that the degeneracy of the skew symmetric bilinear form requires V be even dimensional. Over C, the sp(V ; ω) for different choices of ω are isomorphic. Exercise 1.11. Describe the commutation relations of the orthogonal algebras and symplectic algebra, taking the respective non-degenerate symmetric bilinear form and skew symmetric bilinear form as given in §1.2 in Humphreys’ book, Remark 1.12. There are different real forms of orthogonal and symplectic algebras.

Exercise 1.13. Let pn be the vector space spanned by the diﬀerential operators ∂ pi := ∂i = (= 1, 2, . . . , n) and Jij := xi∂j − xj∂i (i < j) acting on, say, complex ∂xi n valued smooth functions on R . Show that pn forms a Lie algebra with Lie bracket 0 0 0 0 deﬁned by [D,D ] = DD − D D for all D,D ∈ pn. This is the Poincare Lie algebra in n-dimensions. [Our universe is “built” with p4-modules according to Wigner.] QUANTUM GROUPS 3

1.2. Ideals and homomorphisms. Definition 1.14. A ideal I of a Lie algebra g is a subspace such that [I, g] ⊆ I. If I is an ideal of g, then there is the quotient Lie algebra g/I with the Lie bracket defined for any x + I and y + I by [x + I, y + I] = [x, y] + I. eg:ideals Example 1.15. (1). Clearly 0 and g are ideals. (2). Let Tn(C) be the vector space of n × n upper triangular matrices, which is a Lie algebra. Let Nn(C) ⊆ Tn(C) be the subspace of strictly upper triangular matrices (diagonal entries are all equal to 0). Then Nn(C) is an ideal of Tn(C). Exercise 1.16. Let g be a Lie algebra. (1) Show that there is the following sequence g(1) = g, g(2) = [g(1), g(1)], ... , g(n+1) = [g(n), g(n)], ... , of ideals. This is called the derived series of ideals of g. (2) Show that there is the following sequence g1 = g, g2 = [g, g1], ... , gn+1 = [g, gn], ... , of ideals. This is called the descending series of ideals of g. (3) Show that the derived series of ideals of g = Tn(C) terminates, that is, there exists some r such that g(j) = 0 for all j ≥ r. Such Lie algebras are called soluble. (4) Show that the descending series of ideals of Nn(C) terminates. Such Lie algebras are called nilpotent. Definition 1.17. A Lie algebra g is called simple if it is not abelian and contains no ideal other than 0 and g itself. A Lie algebra is called semi-simple if it does not have any non-zero abelian ideal. Remark 1.18. A simple Lie algebra is semi-simple.

Example 1.19. The Lie algebra sl2(C) is simple. In fact, sln(C) are simple for all n ≥ 2. However, the general linear Lie algebra gln(C) is not simple, as the identity matrix In spans an ideal since [In, x] = 0 for all x ∈ gln(C).

Exercise 1.20. If g1 and g2 are Lie algebras, their direct sum, denote by g1 ⊕ g2, is the Lie algebra with underlying vector space g1 ⊕ g2, and Lie bracket [x1 + x2, y1 + y2] = [x1, y1] + [x2, y2] for all x1, y1 ∈ g1 and x2, y2 ∈ g2. Show that if g1 and g2 are both simple, then g1 ⊕ g2 is semi-simple. Deﬁnition 1.21. A homomorphism ϕ : g −→ l of Lie algebras is a linear map such that ϕ([x, y]) = [ϕ(x), ϕ(y)] for all x, y ∈ g. Related notions: injection, surjection, bijection (i.e., isomrphism); ker(ϕ) is an ideal of g; im(ϕ) is a subalgebra of l; im(ϕ) ∼= g/ ker(ϕ). Deﬁnition 1.22. A representation of a Lie algebra g is a homomorphism ϕ : g −→ gl(V ) for some vector space V . Example 1.23. (The adjoint representation) For any Lie algebra L, we a Lie algebra homomorphism ad : L −→ gl(L), x 7→ adx

with adx deﬁned by adx(v) = [x, v] for all v ∈ L. 4 QUANTUM GROUPS

Proof. For any x, y, v ∈ L, we have

adxday(v) = [x, [y, v]], adydax(v) = [y, [x, v]], ad[x,y](v) = [[x, y], v]. By the Jacobi identity, [[x, y], v] = [x, [y, v]] − [y, [x, v]]. Hence

ad[x,y](v) = adxday(v) − adydax(v) = [adx, day](v), ∀v ∈ L. This shows that ad is a Lie algebra homomorphism. Exercise 1.24. Show that the adjoint representation of a semi-simple Lie algebra is injective. Deduce that any semi-simple Lie algebra is a linear Lie algebra in the sense that it is isomorphic to a subalgebra of a general linear Lie algebra. The Killing form κ : g × g −→ g of a ﬁnite dimensional Lie algebra g is the bilinear form deﬁned by

κ(x, y) = tr(adxady), ∀x, y ∈ g. Lemma 1.25. The Killing form of a Lie algebra g is invariant in the sense that κ([x, y], z) = κ(x, [y, z]), x, y, z ∈ g. Hence its radical Rad(κ) := {x ∈ g | κ(x, y) = 0, ∀y ∈ g} is an ideal. Proof. For any x, y, z ∈ g, we have

κ([x, y], z) =tr(ad[x,y]adz) = tr((adxady − adyadx)adz)

=tr(adxadyadz − adxadz) = tr(adxad[y,z]) = κ(x, [yz]). This proves the invariance of κ. If x ∈ Rad(κ), then for all y, z ∈ g, κ([x, y], z) = κ(x, [y, z]) = 0, that is, [x, y] ∈ Rad(κ) for all y ∈ g, hence Rad(κ) is an ideal of g. Theorem 1.26 (Cartan’s criterion). A non-zero ﬁnite dimensional Lie algebra g is semi-simple if and only if its Killing form is non-degenerate. Proof. Assume that I is an abelian ideal of g. Then for any x ∈ I, we have

adxady(v) = [x, [y, v]] ∈ I ∀y, v ∈ g.

Thus tr(adxady) is equal to the trace of the restriction of adxady to I. However, [x, [y, v]] = 0 if v ∈ I. Hence tr(adxady) = 0 for all y ∈ g. Write S = Rad(κ), then I ⊆ S. Therefore, if the Killing form is non-degenerate, g is semi-simple. To prove the opposite direction, we let ad(S) = {adx | x ∈ S} (adx is deﬁned for g). Then ad(S) is a Lie subalgebra of gl(V ) with V = g, and it obviously satisﬁes the condition of Exercise 1.27 below. Hence ad(S) is soluble, and thus S is soluble. Therefore, S is a soluble ideal of g, which must be 0 if g is semi-simple. ex:Cartan Exercise 1.27. Let L be a Lie subalgebra of gl(V ), where dim V < ∞. Suppose that tr(xy) = 0 for all x ∈ [L, L], y ∈ L, then L is soluble. Remark 1.28. It is known that any semi-simple Lie algebra is a direct sum of simple ideals. See, e.g., §5.1 in Humphreys’ book, for the proof.