Finite Dimensional Algebras and Quantum Groups

The interplay between finite dimensional algebras and Lie theory dates back many years. In more recent times, these interrelations have become even more strikingly apparent. This text combines, for the first time in book form, the theories of finite dimensional algebras and quantum groups. More precisely, it investigates the Ringel-Hall algebra realization for the positive part of a quantum enveloping algebra associated with a symmetrizable Cartan matrix and it looks closely at the Beilinson-Lusztig-MacPherson realization for the entire quantum $\mathfrak {gl}_n$. The book begins with the two realizations of generalized Cartan matrices, namely, the graph realization and the root datum realization. From there, it develops the representation theory of quivers with automorphisms and the theory of quantum enveloping algebras associated with Kac-Moody Lie algebras.These two independent theories eventually meet in Part 4, under the umbrella of Ringel-Hall algebras. Cartan matrices can also be used to define an important class of groups - Coxeter groups - and their associated Hecke algebras. Hecke algebras associated with symmetric groups give rise to an interesting class of quasi-hereditary algebras, the quantum Schur algebras. The structure of these finite dimensional algebras is used in Part 5 to build the entire quantum $\mathfrak{gl}_n$ through a completion process of a limit algebra (the Beilinson-Lusztig-MacPherson algebra). The book is suitable for advanced graduate students. Each chapter concludes with a series of exercises, ranging from the routine to sketches of proofs of recent results from the current literature.

Table of Contents:
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 $\pm$-part: Ringel-Hall algebras Bases of quantum enveloping algebras of finite type Green's theorem The BLM algebra: A realization for quantum $\mathfrak{gl}_{n}$: Serre relations in quantum Schur algebras Constructing quantum $\mathfrak{gl}_{n}$ 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.