A Journey Through Representation Theory: From Finite Groups to Quivers via Algebras

Caroline / Serganova Gruson - Collection Yellow Sale 2024

223 pages, parution le 25/10/2018

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Résumé

This text covers a variety of topics in representation theory and is intended for graduate students and more advanced researchers who are interested in the field. The text includes in particular infinite-dimensional unitary representations for abelian groups, Heisenberg groups and SL(2), and representation theory of finite-dimensional algebras.

This text covers a variety of topics in representation theory and is intended for graduate students and more advanced researchers who are interested in the field.

The book begins with classical representation theory of finite groups over complex numbers and ends with results on representation theory of quivers. The text includes in particular infinite-dimensional unitary representations for abelian groups, Heisenberg groups and SL(2), and representation theory of finite-dimensional algebras. The last chapter is devoted to some applications of quivers, including Harish-Chandra modules for SL(2). Ample examples are provided and some are revisited with a different approach when new methods are introduced, leading to deeper results. Exercises are spread throughout each chapter.

Prerequisites include an advanced course in linear algebra that covers Jordan normal forms and tensor products as well as basic results on groups and rings.

Introduction to Representation Theory of Finite Groups.- Modules with Applications to Finite Groups.- Representations of Compact Groups.- Results About Unitary Representations.- On Algebraic Methods.- Symmetric Groups, Schur-Weyl Duality and Positive Self-adjoint Hopf Algebras.- Introduction to representation theory of quivers.- Representations of Dynkin and affine quivers.- Applications of quivers.- Bibliography.- Index. Caroline Gruson is Professor of Mathematics at Universite de Lorraine.

Vera Serganova is Professor of Mathematics at University of California, Berkeley.