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 preference to geometric, method of affine quantum Schur-Weyl idea. to start, quite a few 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 contains 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**

**Example text**

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.

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 . Take an arbitrary a ∈ NI . We proceed by induction on σ (a) = i∈I ai to show u a ∈ H . If σ (a) = 0 or 1, then clearly u a ∈ H . Now let σ (a) > 1. 1), u a ∈ H . So we may assume a is sincere. The case where a1 = · · · = an is trivial. Suppose now there exists i ∈ I such that ai = ai+1 . Define a = (a j ), a = (a j ) ∈ NI by aj = ai − 1, if j = i ; ai , otherwise, and a j = ai +1 − 1, if j = i + 1; ai , otherwise.

Hence, is an isomorphism. Similarly, − is an isomorphism, too. Let K be the ideal of U(C∞ ) generated by k±1 s −1 for s ∈ J∞ . It is clear that K ⊆ Ker . The triangular decomposition U(C∞ ) = U+ ⊗ U0 ⊗ U− implies that U(C∞ )/K = U+ ⊗ Q(v)[K 1±1 , . . , K n±1 ] ⊗ U− + K /K. Since D (n) = D (n)+ ⊗D (n)0 ⊗D (n)− = D (n)+ ⊗Q(v)[K 1±1, . . , K n±1 ]⊗D (n)−, we conclude that Ker = K, as required. Finally, it is straightforward to check that K is a Hopf ideal. 5. The double Ringel–Hall algebra D (n) is a Hopf algebra with comultiplication , counit ε, and antipode σ defined by (E i ) = E i ⊗ K i + 1 ⊗ E i , (K i±1 ) = K i±1 ⊗ K i±1 , (Fi ) = Fi ⊗ 1 + K i−1 ⊗ Fi , ± ± (z± s ) = zs ⊗ 1 + 1 ⊗ zs ; ε(E i ) = ε(Fi ) = 0 = ε(z± s ), σ (E i ) = −E i K i−1 , ε(K i ) = 1; σ (Fi ) = − K i Fi , σ (K i±1 ) = K i∓1 , ± and σ (z± s ) = −zs , where i ∈ I and s ∈ J∞ .