By Bangming Deng

ISBN-10: 1607092050

ISBN-13: 9781607092056

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Additional info for A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory

Example text

If we denote the standard basis of X by e1 , . . , en and the dual basis by f 1 , . . , f n , then f i , e j rd = δi, j . 1) with en+1 = e1 and f n+1 = f 1 define a root datum (Y, X, , rd , . ). For notational simplicity, we shall identify both X and Y with ZI by setting ei = i = f i for all i ∈ I . Under this identification, the form , rd : ZI × ZI → Z becomes a symmetric bilinear form, which is different from the Euler forms , and its symmetrization ( , ). However, they are related as follows.

More precisely, we will modify the construction in [40] to extend Beck’s algebra embedding of the quantum affine sln into the quantum loop algebra U(gln ) defined by Drinfeld’s new presentation [20] to an explicit isomorphism from D (n) to U(gln ). 1, consider the loop algebra gln = gln ⊗ Q[t, t −1 ], 50 2. Double Ringel–Hall algebras of cyclic quivers which is generated by E i,i+1 ⊗ t s , E i +1,i ⊗ t s , and E j, j ⊗ t s , for s ∈ Z, 1 i n − 1, and 1 j n. We have the following quantum enveloping algebra associated with gln ; see [20] (and also [28, 40]).

This implies in particular that both U + and U − are free Z-modules. We now look at the structure of U 0 . Let V 0 be the Z-subalgebra generated ks ;0 by K i±1 , K it ;0 , k±1 0, and s 1. 4. Some integral forms 47 that V 0 is Z-free. 14] (see also [22] and [12, Th. 20]) shows that V 0 contains the Z-subalgebra V 1 generated by K i±1 and K it ;0 which has a Z-basis n a K= Ki i i=1 n−1 a K = Ki i i=1 Ki ; 0 ti ai ∈ {0, 1}, ti ∈ N, ∀i ∈ I , Ki ; 0 Kn ; 0 · K ian ti tn ai ∈ {0, 1}, ti ∈ N, ∀i ∈ I .