A Double Hall Algebra Approach to Affine Quantum Schur-Weyl by Bangming Deng
By Bangming Deng
The idea of Schur-Weyl duality has had a profound impact over many parts of algebra and combinatorics. this article is unique in respects: it discusses affine q-Schur algebras and offers an algebraic, rather than geometric, method of affine quantum Schur-Weyl idea. to start, a number of algebraic constructions are mentioned, together with double Ringel-Hall algebras of cyclic quivers and their quantum loop algebra interpretation. the remainder of the e-book 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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Extra info for A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory
3) The proof above can be easily modified to construct a C-algebra isomorphism EH,C : D ,C (n) → UC (gln ), where the algebras are defined over C with respect to a non-root-of-unity z ∈ C∗ .
1) is defined over A. Hence, we can form a double Ringel–Hall algebra D (n)A . Then D ,C (n) ∼ = D (n)A ⊗ C and D (n) = D (n)A ⊗ Q(v). 2. Schiffmann–Hubery generators In this and the following sections, we will investigate the structure of D (n) by relating it with the quantum enveloping algebra of a generalized Kac–Moody algebra based on [67, 39]; see also [38, 14]. 1). Recall that an element of a Hopf algebra with comultiplication is called primitive if 38 2. Double Ringel–Hall algebras of cyclic quivers (x) = x ⊗ 1 + 1 ⊗ x.
Also, let U(gln )0 be i,0 , εn , θ−s ), for 1 the subalgebra of U(gln ) generated by the ki±1 . 1) induces that of U(gln ). 4. The multiplication map U(gln )+ ⊗ U(gln )0 ⊗ U(gln )− −→ U(gln ) is a Q(v)-space isomorphism. 5. 12)]. Indeed, under the isomorphism EH , C is identified with the central subalgebra Z (n) of D (n). (2) In  a Hopf algebra isomorphism H (n) 0 → U(gln )+ ⊗ U(gln )0 −1 was established and, moreover, the elements EH−1 (x+j,−1 k−1 j ) and EH (gi,±s ) in D (n) were explicitly described.