By Bangming Deng

The speculation of Schur-Weyl duality has had a profound effect over many parts of algebra and combinatorics. this article is unique in respects: it discusses affine q-Schur algebras and provides an algebraic, instead of geometric, method of affine quantum Schur-Weyl conception. to start, a variety of algebraic buildings are mentioned, together with double Ringel-Hall algebras of cyclic quivers and their quantum loop algebra interpretation. the remainder of the publication investigates the affine quantum Schur-Weyl duality on 3 degrees. This comprises the affine quantum Schur-Weyl reciprocity, the bridging position of affine q-Schur algebras among representations of the quantum loop algebras and people of the corresponding affine Hecke algebras, presentation of affine quantum Schur algebras and the realisation conjecture for the double Ringel-Hall algebra with an explanation of the classical case. this article is perfect for researchers in algebra and graduate scholars who are looking to grasp Ringel-Hall algebras and Schur-Weyl duality.

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**Extra resources for A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory**

**Example text**

If a is not sincere, say ai = 0, then ua = j∈I, j =i v a j (1−a j ) ai−1 ai+1 u i−1 · · · u a11 u ann · · · u i+1 ∈ C (n). [a j ]! 1) Thus, H (n) is generated by u i and u a , for i ∈ I and sincere a ∈ NI . Indeed, this result can be strengthened as follows; see also [67, p. 421]. 3. The Ringel–Hall algebra H (n) is generated by u i and u mδ , for i ∈ I and m 1. Proof. Let H be the Q(v)-subalgebra generated by u i and u mδ for i ∈ I and m 1. To show H = H (n), it suffices to prove u a = u [Sa ] ∈ H for all a ∈ NI .

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. 2. For a = then (1) (a, b) = a˜ , b rd ai i ∈ ZI , if we put a˜ = ; (2) a, b = a , b ai i˜ and a = rd ai i , , for all a, b ∈ ZI.

For m 0, applying Lusztig’s formulas km+2 s ks ; 0 t = v t (v − v −1 )km+1 s k−m−1 s ks ; 0 t = −v −t (v − v −1 )k−m s ks ;0 in [52, p. 278] yields k±m s t 2 therefore, W2 = W1 = V . ks ; 0 t +1 + v 2t km s ks ; 0 t +1 ∈ W1 , for any m ks ; 0 t + v −2t k−m+1 s and ks ; 0 t 0. Hence, W2 ⊆ W1 and, 48 2. 3. The integral 0-part U 0 is the Z-subalgebra of U(C∞ ) genks ;0 erated by K i±1 , K it ;0 , k±1 1. Hence, U 0 is a free s , 1 , for i ∈ I and s Z-module with basis {x y | x ∈ K, y ∈ M}. Proof. Let U = U + V 0 U − .