By Bangming Deng
The speculation 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 offers an algebraic, instead of geometric, method of affine quantum Schur-Weyl idea. to start, numerous algebraic constructions are mentioned, together with double Ringel-Hall algebras of cyclic quivers and their quantum loop algebra interpretation. the remainder of the ebook 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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Extra resources for A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory
Preliminaries which play an important role in the theory of quantum groups. We shall fix the following root datum throughout the book. 1. Let X = Zn , Y = Hom(X, Z), and let , rd : Y × X → Z be the natural perfect pairing. If we denote the standard basis of X by e1 , . . , 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 .
The following lemma can be checked directly. 2. Let A = (A , μA , ηA , A , εA , σA ), B = (B, μB , ηB , B , εB , σB ) be two Hopf algebras over a field F together with a skewHopf pairing ψ : A × B → F. Assume that σA and σB are both invertible. , op op −1 −1 (A , μA , ηA , A , εA , σA ) and (B, μB , ηB , B , εB , σB )). 1. 5. First, we need a skew-Hopf pairing. 2 to [78, Prop. 3] yields the following result. For completeness, we sketch a proof. We introduce some notation which is used in the proof.
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. 1) A where the sum is taken over all A ∈ + (n) such that d(A) = mδ and soc M(A) is square-free. We also define c0± = 1 by convention.