Download e-book for iPad: A Double Hall Algebra Approach to Affine Quantum Schur-Weyl by Bangming Deng

By Bangming Deng

ISBN-10: 1607092050

ISBN-13: 9781607092056

The idea of Schur-Weyl duality has had a profound impact over many components of algebra and combinatorics. this article is unique in respects: it discusses affine q-Schur algebras and provides an algebraic, instead of geometric, method of affine quantum Schur-Weyl conception. to start, a variety of algebraic constructions 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 explanation 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.

Show description

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

Best algebra & trigonometry books

Get Abels Beweis German PDF

Aus den Rezensionen zur englischen Auflage: "Die Leser von Pesics faszinierendem kleinen Buch werden zu dem unausweichlichen Urteil kommen: Niels [Henrik] Abel hat sich der Genialität im fünften Grade schuldig gemacht. " William Dunham, Muhlenberg collage und Autor von "Journey via Genius: the nice Theorems of arithmetic "Peter Pesic schreibt über Abels Werk mit Begeisterung und Einfühlungsvermögen, und ruft Erinnerungen an die großartigen Momente in der Entwicklung der Algebra wach.

Download PDF by James E. Humphreys: Representations of Semisimple Lie Algebras in the BGG

This is often the 1st textbook therapy of labor resulting in the landmark 1979 Kazhdan-Lusztig Conjecture on characters of easy maximum weight modules for a semisimple Lie algebra $\mathfrak{g}$ over $\mathbb {C}$. The environment is the module type $\mathscr {O}$ brought by way of Bernstein-Gelfand-Gelfand, such as all optimum weight modules for $\mathfrak{g}$ similar to Verma modules and finite dimensional uncomplicated modules.

Download e-book for iPad: Noncommutative Rational Series with Applications by Jean Berstel

The algebraic concept of automata was once created through Sch? tzenberger and Chomsky over 50 years in the past and there has seeing that been loads of improvement. Classical paintings at the thought to noncommutative strength sequence has been augmented extra lately to components similar to illustration conception, combinatorial arithmetic and theoretical laptop technology.

Additional resources for A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory

Example text

Schiffmann–Hubery generators 37 where I denotes the ideal generated by 1 ⊗ K α − K α ⊗ 1, for all α ∈ ZI . By the construction, I is indeed a Hopf ideal of D (n). Thus, D (n) is again a Hopf algebra. We call D (n) the double Ringel–Hall algebra of the cyclic quiver (n). , D (n)− ) be the Q(v)-subalgebra of D (n) generated − + by u + (n). , u A ) for all A ∈ D (n) generated by K α for all α ∈ ZI . Then D (n)+ ∼ = H (n), D (n)− ∼ = H (n)op , and D (n)0 ∼ = Q(v)[K 1±1 , . . , K n±1 ]. 3) Moreover, the multiplication map D (n)+ ⊗ D (n)0 ⊗ D (n)− −→ D (n) is an isomorphism of Q(v)-vector spaces.

0 0 0 · · · v ±(n−2)s −v ±ns ⎠ 1 1 1 1 1 ∓ [s] ∓ [s] ∓ [s] ··· ∓ [s] ∓ [s] By the definition of hi,±s and θ±s , n hi,±s = n (±s) X i, j g j,±s , for 1 i < n, and θ±s = j =1 (±s) X n, j g j,±s . j =1 A direct calculation shows that det(X (±s) ) = ∓ 1 1 + v ±2s + · · · + v ±2(n−1)s [s] n−2 v ±is = 0 (n (±s) We denote the inverse of X (±s) by Y (±s) = (Yi, j ). Thus, for each 1 n−1 gi,±s = (±s) 2). i=1 i n, (±s) Yi, j h j,±s + Yi,n θ±s . j =1 Therefore, the Q(v)-subspace of U(gln ) spanned by g1,±s , .

5. The double Ringel–Hall algebra D (n) is a Hopf algebra with comultiplication , counit ε, and antipode σ defined by (E i ) = E i ⊗ K i + 1 ⊗ E i , (K i±1 ) = K i±1 ⊗ K i±1 , (Fi ) = Fi ⊗ 1 + K i−1 ⊗ Fi , ± ± (z± s ) = zs ⊗ 1 + 1 ⊗ zs ; ε(E i ) = ε(Fi ) = 0 = ε(z± s ), σ (E i ) = −E i K i−1 , ε(K i ) = 1; σ (Fi ) = − K i Fi , σ (K i±1 ) = K i∓1 , ± and σ (z± s ) = −zs , where i ∈ I and s ∈ J∞ . 6. (1) For notational simplicity, we sometimes continue to use u i± as generators of D (n). 4. Some integral forms for i ∈ I and s 45 1.

Download PDF sample

A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory by Bangming Deng


by Donald
4.2

Rated 4.75 of 5 – based on 46 votes