Bangming Deng

Finite dimensional algebras and quantum groups

Getting started Quivers and their representations: Representations of quivers Algebras with Frobenius morphisms Quivers with automorphisms Some quantized algebras: Coxeter groups and Hecke algebras Hopf algebras and universal enveloping algebras Quantum enveloping algebras Representations of symmetric groups: Kazhdan-Lusztig combinatorics for Hecke algebras Cells and representations of symmetric groups The integral theory of quantum Schur algebras Ringel-Hall algebras: A realization for the

Generic Extensions and Canonical Bases for Cyclic Quivers

\pm

A double hall algebra approach to affine quantum Schur-Weyl theory

-part: Ringel-Hall algebras Bases of quantum enveloping algebras of finite type Greens theorem The BLM algebra: A realization for quantum

QUASI-HEREDITARY ALGEBRAS ASSOCIATED WITH UPPER TRIANGULAR BIMODULES

\mathfrak{gl}_{n}

Linear quivers, generic extensions and Kashiwara operators

: Serre relations in quantum Schur algebras Constructing quantum

A necessary condition for the finiteness of Δ-good module categories

\mathfrak{gl}_{n}

A Double Hall Algebra Approach to Affine Quantum Schur–Weyl Theory: Bibliography

via quantum Schur algebras Appendices: Varieties and affine algebraic groups Quantum linear groups through coordinate algebras Quasi-hereditary and cellular algebras Bibliography Index of notation Index of terminology.

A Double Hall Algebra Approach to Affine Quantum Schur–Weyl Theory: Index

We use the monomial basis theory developed by Deng and Du to present an elementary al- gebraic construction of the canonical bases for both the Ringel-Hall algebra of a cyclic quiver and the positive part U + of the quantum affine sln. This construction relies on analysis of quiver representa- tions and the introduction of a new integral PBW-like basis for the Lusztig Z(v, v 1 )-form of U + .

A Double Hall Algebra Approach to Affine Quantum Schur–Weyl Theory: Contents

Introduction 1. Preliminaries 2. Double Ringel-Hall algebras of cyclic quivers 3. Affine quantum Schur algebras and the Schur-Weyl reciprocity 4. Representations of affine quantum Schur algebras 5. The presentation and realization problems 6. The classical (v =1) case Bibliography Index.

A Double Hall Algebra Approach to Affine Quantum Schur–Weyl Theory: Frontmatter

Let A be an aggregate with a finite spectroid S and B a bimodule over A. If B is upper triangular, it is shown by Brüstle and Hille that the category mat B of matrices over B is equivalent to the Δ-good module category over a quasi-hereditary algebra which is the opposite of the endomorphism algebra of a projective generator in mat B. In the present paper we provide an explicit construction of indecomposable projectives and injectives in mat B by defining left and right radicals of B. In particular, we obtain a description of the characteristic tilting module over the quasi-hereditary algebra associated with B. Moreover, the Ringel dual of this quasi-hereditary algebra is the opposite of the quasi-hereditary algebra associated with a bimodule over A op.