The theory of Schur–Weyl duality has had a profound influence over many areas of algebra and combinatorics. This text is original in two respects: it discusses affine q-Schur algebras and presents an algebraic, as opposed to geometric, approach to affine quantum Schur–Weyl theory. To begin, various algebraic structures are discussed, including double Ringel–Hall algebras of cyclic quivers and their quantum loop algebra interpretation. The rest of the book investigates the affine quantum Schur–Weyl duality on three levels. This includes the affine quantum Schur–Weyl reciprocity, the bridging role of affine q-Schur algebras between representations of the quantum loop algebras and those of the corresponding affine Hecke algebras, presentation of affine quantum Schur algebras and the realisation conjecture for the double Ringel–Hall algebra with a proof of the classical case. This text is ideal for researchers in algebra and graduate students who want to master Ringel–Hall algebras and Schur–Weyl duality.

We use the Hecke algebras of affine symmetric groups and their associated Schur algebras to construct a new algebra through a basis, and a set of generators and explicit multiplication formulas of basis elements by generators. We prove that this algebra is isomorphic to the quantum enveloping algebra of the loop algebra of gl n . Though this construction is motivated by the work [1] by Beilinson-Lusztig-MacPherson for quantum gl n , our approach is purely algebraic and combinatorial, independent of the geometric method which seems to work only for quantum gl n and quantum affine sln. As an application, we discover a presentation of the Ringel-Hall algebra of a cyclic quiver by semisimple generators and their multiplications by the defining basis elements.The double Ringel-Hall algebra construction of quantum affine gl n is an affine generalisation of a similar construction for a quantum enveloping algebra U of a finite type quiver via a Ringel-Hall algebra which, as the positive or negative part of U, is spanned by the basis of isoclasses of representations of the quiver and whose multiplication is defined by Hall polynomials, see [15,16,20]. However, there is another construction for quantum gl n by Beilinson, Lusztig and MacPherson [1, 5.7], which directly displays a basis for the entire quantum enveloping algebra U(gl n ) and displays the multiplication rules by explicit formulas of basis elements by generators.

We will construct the Lusztig form for the quantum loop algebra of gl n by proving the conjecture [4, 3.8.6] and establish partially the Schur-Weyl duality at the integral level in this case. We will also investigate the integral form of the modified quantum affine gl n by introducing an affine stabilisation property and will lift the canonical bases from affine quantum Schur algebras to a canonical basis for this integral form. As an application of our theory, we will also discuss the integral form of the modified extended quantum affine sln and construct its canonical basis to verify a conjecture of Lusztig in this case.

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We introduce a spanning set of Beilinson-Lusztig-MacPherson type, {A(j, r )} A,j , for affine quantum Schur algebras S (n, r ) and construct a linearly independent set {A(j)} A,j for an associated algebra K (n). We then establish explicitly some multiplication formulas of simple generators E h,h+1 (0) by an arbitrary element A(j) in K (n) via the corresponding formulas in S (n, r ), and compare these formulas with the multiplication formulas between a simple module and an arbitrary module in the Ringel-Hall algebras H (n) associated with cyclic quivers. This allows us to use the triangular relation between monomial and PBW type bases for H (n) established in Deng and Du (Adv Math 191:276-304, 2005) to derive similar triangular relations for S (n, r ) and K (n). Using these relations, we then show that the subspace A (n) of K (n) spanned by {A(j)} A,j contains the quantum enveloping algebra U (n) of affine type A as a subalgebra. As an application, we prove that, when this construction is applied to quantum Schur algebras S(n, r ), the resulting subspace A(n) is in fact a subalgebra which is isomorphic to the quantum enveloping algebra of gl n . We conjecture that A (n) is a subalgebra of K (n).

In [J. Du, Q. Fu, J.-P. Wang, Infinitesimal quantum gl n and little q-Schur algebras, J. Algebra 287 (2005) 199-233], little q-Schur algebras were introduced as homomorphic images of infinitesimal quantum gl n in the odd root of unity case. In this paper, we shall introduce little q-Schur algebras u k (n, r) at even roots of unity. We shall construct various bases for u k (n, r) and give the dimension formula of u k (n, r).

In this article, we describe the relation between the ilifinitesimal q-Sclwr algebra defined in Cox (1997) am/ the little q-Schur algebra tlefiued in Du eta/. (2005).Key Words: In fi nites imal q-Schur a lge bra; Littl e q-Schur algebra ; q-Sc hur a lge bra.Mathematics Subject Classification: 17B37; 20G05.

Let D△(n) be the double Ringel-Hall algebra of the cyclic quiver △(n) and letḊ△(n) be the modified quantum affine algebra of D△(n). We will construct an integral formḊ△(n) foṙ D△(n) such that the natural algebra homomorphism fromḊ△(n) to the integral affine quantum Schur algebra is surjective. Furthermore, we will use Hall algebras to construct the integral formand prove that the natural algebra homomorphism from U Z ( gl n ) to the affine Schur algebra over Z is surjective.QIANG FU conjectured in [5, 3.8.6] that D △ (n) is a Z-subalgebra of D △ (n). If this conjecture is true, then D △ (n) becomes an integral form for D △ (n).LetḊ △ (n) be the modified quantum affine algebra of D △ (n). Associated with D △ (n), we will construct a certain free Z-submodule ofḊ △ (n), denoted byḊ △ (n), such thatḊ △ (n) = D △ (n) ⊗ Z Q(υ). We will prove in 4.2 and 4.3 thatḊ △ (n) is a Z-subalgebra ofḊ △ (n) and the natural algebra homomorphismζ r fromḊ △ (n) to S △ (n, r) is surjective, where S △ (n, r) is the affine quantum Schur algebra over Z.