By Bangming Deng

The idea of Schur-Weyl duality has had a profound impression over many parts of algebra and combinatorics. this article is unique in respects: it discusses affine q-Schur algebras and offers an algebraic, in place of geometric, method of affine quantum Schur-Weyl idea. to start, a number 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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K n±1 ]. 1) Putting x = x ⊗1 and y = 1⊗ y for x ∈ H (n) and y ∈ Q(v)[K 1±1 , . . , K n±1 ], + H (n) 0 is a Q(v)-space with basis {u + (n)}. We are A K α | α ∈ ZI, A ∈ now ready to introduce the Ringel–Green–Xiao Hopf structure on H (n) 0 . Let + + (n)∗ := (n)\{0}. 4. The Q(v)-space H (n) 0 with basis {u + A K α | α ∈ ZI, A ∈ + (n)} becomes a Hopf algebra with the following algebra, coalgebra, and antipode structures. + (a) Multiplication and unit (Ringel [64]): for all A, B ∈ β ∈ ZI , + u+ AuB = v C∈ Kα u+ A =v + d( A),d(B) (n) and α, ϕ CA,B u + C, (n) d( A),α u+ A Kα , K α K β = K α+β , and 1 = u+ 0 = K0.

Note that we have assumed here that σB is invertible. Let A ∗ B be the free product of F-algebras A and B with identity. Then A ∗ B is the coproduct of A and B in the category of F-algebras. More precisely, for any fixed bases BA and BB for A , B, respectively, where both BA and BB contain the identity element, A ∗ B is the F-vector space spanned by the basis consisting of all words b1 b2 · · · bm (bi ∈ (BA \{1}) ∪ (BB \{1})) of any length m 0 such that bi bi +1 is not defined (in other words, bi , bi+1 are not in the same A or B) with multiplication given by “contracted juxtaposition” ⎧ ⎪ b · · · bm b1 · · · bm , if bm b1 is not defined; ⎪ ⎪ 1 ⎪ ⎨b · · · b if bm b1 is defined, 1 m−1 cb2 · · · bm , (b1 · · · bm ) ∗ (b1 · · · bm ) = ⎪ bm b1 = 0; ⎪ ⎪ ⎪ ⎩ 0, otherwise.

The Q(v)-space H (n) 0 with basis {u + A K α | α ∈ ZI, A ∈ + (n)} becomes a Hopf algebra with the following algebra, coalgebra, and antipode structures. + (a) Multiplication and unit (Ringel [64]): for all A, B ∈ β ∈ ZI , + u+ AuB = v C∈ Kα u+ A =v + d( A),d(B) (n) and α, ϕ CA,B u + C, (n) d( A),α u+ A Kα , K α K β = K α+β , and 1 = u+ 0 = K0. (b) Comultiplication and counit (Green [34]): for all C ∈ (u + C) = v A,B∈ + d( A),d(B) (n) + (n) and α ∈ ZI , aAaB C + ϕ u ⊗ u+ A K d(B) , aC A,B B (K α ) = K α ⊗ K α , ε(u + C ) = 0 (C = 0), and ε(K α ) = 1.