By Bangming Deng

The idea of Schur-Weyl duality has had a profound effect over many components of algebra and combinatorics. this article is unique in respects: it discusses affine q-Schur algebras and provides an algebraic, in place 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 function 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 evidence 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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**Additional info for A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory**

**Sample text**

Ys , k±1 s ), for all s ∈ Jm . Then, by [48, Th. 23], U(C m ) admits a triangular decomposition U(C m ) = U+ ⊗ U0 ⊗ U− . 3. Also, we denote by U (n) the subalgebra of U(Cm ) generated by E i , Fi , K i , K i−1 (i ∈ I ). 1 follows immediately from the following result which describes the structure of D (n). It is a generalization of a result for the double Ringel–Hall algebra of a finite dimensional tame hereditary algebra in [38] to the cyclic quiver case. 1. 4. There is a unique surjective Hopf algebra homomorphism : U(C∞ ) → D (n) satisfying, for each i ∈ I and s ∈ J∞ , − ±1 − E i −→ u i+ , xs −→ z+ −→ K i± , k± s , Fi −→ u i , ys −→ zs , K i s −→ 1.

3(g8)] with t = 1 becomes ks ; 0 t vt ks ; 0 ks ; 0 − [t]ks 1 t ks ; 0 . 2. The set δ M := k11 · · · kδmm k1 ; 0 t1 ··· km ; 0 tm 1, δi ∈ {0, 1}, ti ∈ N m forms a Z-basis of V 2 . Proof. Let W1 = spanZ M and W2 = spanZ kδ11 · · · kδmm k1 ; 0 t1 ··· km ; 0 tm m 1, δi ∈ Z, ti ∈ N . 3), kst;0 ∈ V 2 , for s 1 and t 0. Thus W1 ⊆ W2 ⊆ V 2 . Clearly, ks ;0 2 k±1 s W2 ⊆ W2 . 3) again, 1 W2 ⊆ W2 . Thus, V W2 ⊆ W2 . Since 2 2 1 ∈ W2 , V ⊆ W2 and, hence, V = W2 . 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.

3. 4, there are isomorphisms ∼ ∼ C (n)+ −→ U(sln )+ , u i+ −→ E i , and C (n)− −→ U(sln )− , u i− −→ Fi . Here we have applied the anti-involution U(sln )+ → U(sln )− , E i → Fi . 3), there are decompositions D (n)+ = C (n)+ ⊗Q(v) Z (n)+ where Z (n)± m 1. 1) ± := Q(v)[z± , z , . 3. 4. 2 can be constructed for D ,C (n)± . In particular, we obtain decompositions D ± ,C (n) ± ± = C (n)± C ⊗ C[z1 , z2 , . 3. 3. 4, we describe a presentation for D (n) as follows. 1. 5 that D (n) can be obtained as a specialization from the ,C Z-algebra D (n) by the base change Z → C, v → z.