based on the talk by Markus Reineke (Wuppertal) January 15, 2002

Introduction to quantum groups and crystal bases

based on the talk by Markus Reineke (Wuppertal) January 15, 2002

Let g be a finite dimensional complex Lie algebra. Examples of Lie alge- bras are gl_nand sl_n, n ≥ 2. We will always assume that g is a semisimple Lie algebra, i.e. g =Lk

i=1g_i, where g_i, i = 1, . . . , k, is a simple Lie algebra, that is [−, −] 6= 0 and for each I ⊂ g_i such that [g_i, I] ⊂ I we have either I = 0 or I = g_i. The semisimple Lie algebras are classified by Dynkin diagrams or, equivalently, by Cartan matrices. For example, sl_n is a simple Lie algebra of type An−1 and sl₃ corresponds to the matrix _{−1 2}^{2 −1}
.

The representation of a Lie algebra g in a vector space V is a Lie algebra homomorphism g → gl(V ). Weyl showed that if g is semisimple then the category mod g of finite dimensional representations of g is semisimple. Let U (g) be the universal enveloping algebra of g. The categories mod g and mod U (g) are equivalent, thus the category mod U (g) is semisimple.

Recall that a complex Lie algebra g has a decomposition g = n⁻⊕ h ⊕ n⁺. For example if g = sl_n, then n⁻ consists of the lower triangular matrices, h consists of the diagonal matrices and n⁺ consists of the upper triangular matrices. We have the generators F_i of n⁻, H_i of h, E_i for n⁺, i ∈ I, where I is the set of vertices of the corresponding Dynkin diagram. If g = sl_n then F_i = e_i+1,i, H_i = e_i,i − e_i+1,i+1 and E_i = e_i,i+1, i = 1, . . . , n − 1. As the consequence U (n⁺) is generated by E_i, i ∈ I, as an algebra.

Let λ = (λ_i)_i∈I ∈ NI, Iλ := P

i∈IU (n⁺)E_i^λi+1 and L_λ := U (n⁺)/I_λ. We define the action of U (g) on L_λ via F_i1 = 0 and H_i1 = λ_i1, i ∈ I. It follows that L_λ, λ ∈ NI, form the complete set of simple U (g)-modules.

An interesting problem connected with the above description is the ques- tion about dim L_λ. Another one is the description of the restriction of L_λ to U (h) = C[Hi | i ∈ I]. This is answered by Weyl character formula, which says that ch L_λ := P

µ∈NIdim(L_λ)_µe^µ =

P

w∈Wsgn(w)e^w(λ+ρ) P

w∈Wsgn(w)e^w(ρ) . However, there is still a question whether there is a “combinatorial formula” for ch L_λ, i.e.

1

a formula of the form (dim L_λ)_µ equals the number of certain combinatorial objects.

We know that L_λ ⊗ L_µ = L

ν∈NIL^c

ν

νλµ for some c^ν_λµ. We may ask how to compute c^ν_λµ. For type A the answer is contained in the Littlewood–

Richardson rule.

We want to deform U (g). However, complex semisimple Lie algebras are rigid, that is all deformations are trivial. Consequently, U (g) is rigid as a cocommutative Hopf algebra. Happily, U (g) is not rigid as a non- cocommutative Hopf algebra. From now on we will assume that g is of one of the types A, D or E6, E7, E8.

Theorem (Serre). We have U (n⁺) = ChEi | E_i ∈ Ii/([E_i, E_j]) = 0 if a_ij = 0, and [E_i, [E_i, E_j]] = 0 if a_ij = −1.

We have [Ei, [Ei, Ej]] = E_i²Ej − 2EiEjEi + EjE_i². Thus we may de- fine Uv(n⁺) := C(v)hEi | i ∈ Ii/([E_i, E_j] = 0 if a_ij = 0 and E_i²E_j − (v + v⁻¹)E_iE_jE_i+ E_jE_i² = 0 if a_ij = −1) and U_v(n⁺) is the Z[v, v⁻¹]-subalgebra of Uv(n⁺) generated by E_i⁽ⁿ⁾, i ∈ I, n ∈ N, where Ei⁽ⁿ⁾ := _[n]!¹ E_iⁿ, and [n] := ^v_v−v^n−v−1⁻ⁿ. It follows easily that C1 ⊗_Z[v,v⁻¹_]U_v(n⁺) ' U (n⁺), where Cµ

denotes a 1-dimensional Z[v, v⁻¹]-module with v acting by multiplication by µ.

Let Q be a quiver obtained from the diagram determining g. For d ∈ NI we define R_d := L

α:i→jHom_k(k^di, k^dj) and G_d := Q

i∈IGL(k^di), where k = Fq for some q. Then G_d acts on R_dvia (g_i) ∗ (X_α) := (g_jX_αg⁻¹_i ). We put H (Q) := L_d∈NIC^Gd(R_d), where C^Gd(R_d) denots the space of G_d-invariant complex functions on R_d. The formula (f ∗ g)(X) := q^αP

U ⊂Xg(U )f (X/U ) defines in H (Q) a structure of an associative C-algebra called the Hall al- gebra. We have H (Q) ' C^√q⊗_Z[v,v⁻¹_]U_v(n⁺).

Let Bq(Q) be the set of the characteristic functions of all orbits in all R_d. Then Bq(Q) is a basis of H (Q). There exists a basis B(Q) of Uv(n⁺), which specializes to Bq(Q) for each q. However, for different orientations Q of the diagram determining g the bases B(Q) are different. Let L (Q) :=

Z[v⁻¹]B(Q). If follows that L (Q) = L (Q⁰) if Q and Q⁰ have the same underlying graph. Thus we put L := L (Q). If π : L → L /v⁻¹L is the canonical projection, then π(B(Q)) = π(B(Q⁰)). We call B := π(B(Q)) the crystal basis of L /v⁻¹L .

There exists the unique basisB of Uv(n⁺) such thatB ⊂ L , π(B) = B and b = b for all b ∈B, where Ei = E_i and v := v⁻¹. Th proof of the above fact uses degenerations.

LetBµbe the specialization of B to Cµ⊗_Z[v,v⁻¹_]B and πλ : U (n⁺) → L_λ be the canonical projection.

2

Theorem (Lusztig/Kashiwara). We have that π_λ(B1) \ {0} is a basis of L_λ for all λ ∈ NI.

Proof. Fix i ∈ I and choose an orientation Q such that i is a source in Q.

Then, it follows that B1(Q) ∩ U (n⁺)E_i^λi+1 is a basis of U (n⁺)E_i^λi+1). As the consequenceB1∩U (n⁺)E_i^λi+1 is a basis of U (n⁺)E_i^λi+1for all i HenceB1∩I_λ is a basis of I_λ and the claim follows.

For example we have a basis of sl_n+1, which is parameterized by triangles (a_ij)_{1≤i≤j≤n}, a_ij ∈ N. The corresponding basis of Lλ is parameterized by those (a_ij), which satisfy P

1≤k≤ia_kj −P

1≤k<ia_k,j−1≤ λ_j for i ≤ j.

3