# 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.

**Read Online or Download A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory PDF**

**Best algebra & trigonometry books**

**An Algebraic Introduction to Complex Projective Geometry: Commutative Algebra**

During this advent to commutative algebra, the writer choses a path that leads the reader in the course of the crucial principles, with out getting embroiled in technicalities. he's taking the reader fast to the basics of advanced projective geometry, requiring just a simple wisdom of linear and multilinear algebra and a few hassle-free workforce concept.

**Inequalities : a Mathematical Olympiad approach**

This publication is meant for the Mathematical Olympiad scholars who desire to organize for the examine of inequalities, a subject now of common use at a variety of degrees of mathematical competitions. during this quantity we current either vintage inequalities and the extra priceless inequalities for confronting and fixing optimization difficulties.

This quantity provides the complaints of the foreign convention on ""Recent growth in Algebra"" that was once held on the Korea complicated Institute of technological know-how and know-how (KAIST) and Korea Institute for complex learn (KIAS). It introduced jointly specialists within the box to debate growth in algebra, combinatorics, algebraic geometry and quantity idea.

- Hereditary Noetherian prime rings and idealizers
- Recent Developments in the Inverse Galois Problem: A Joint Summer Research Conference on Recent Developments in the Inverse Galols Problem July 17-23, ... Seattle
- Galois Theory (Second Edition)
- Introduction to Operator Algebras

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

**Sample text**

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 [40] 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.