Algebra - Representation Theory

Over the last three decades representation theory of groups, Lie algebras and associative algebras has undergone a rapid development through the powerful tool of almost split sequences and the Auslander-Reiten quiver. Further insight into the homology of finite groups has illuminated their representation theory. The study of Hopf algebras and non-commutative geometry is another new branch of representation theory which pushes the classical theory further. All this can only be seen in connection with an understanding of the structure of special classes of rings. The aim of this book is to introduce the reader to some modern developments in:

1. Lie algebras, quantum groups, Hopf algebras and algebraic groups;
2. non-commutative algebraic geometry;
3. representation theory of finite groups and cohomology;
4. the structure of special classes of rings.

Tilting modules for algebraic and quantum groups.- Connections between group representations and cohomology.- Invariant theory of algebra representations.- Representation Theory of Orders.- On some problems in Pi-theory in characteristic p.- Blocks of category O, double centralizer properties, and Enright’s completions.- The Gelfand-Kirillov dimensions of Algebras arising from Representation Theory.- The normalizer of a finite group in its integral group ring and ?ech cohomology.- Representation Theory of Semisimple Hopf Algebras.- Semigroup cohomology and applications.- In Search For Noetherian Algebras.- Cohen-Macaulay representation.- Hereditary abelian categories and almost split sequences.- The abelian defect group conjecture: some recent progress.- 2-dimensional Orders and integral Hecke orders.- Invertible ideals and non-commutative generalizations of regular rings.- Langlands’philosophy and Koszul duality.- Noncommutative projective geometry.- Braid groups as self-equivalencesof derived categories.- Modules with good filtration and invariant theory.