Title: Algebras, representations and homological dimensions
Abstract:This is a series of three lectures for graduate students on representations and homological dimensions of flnite-dimensional algebras. 1. Algebras and their presentations. The flrst lecture will be el...This is a series of three lectures for graduate students on representations and homological dimensions of flnite-dimensional algebras. 1. Algebras and their presentations. The flrst lecture will be elementary. In this part, we shall represent our algebras as path algebras or their quotients, or as diagram algebras. The main result is Gabriel’s Theorem: Every flnite-dimensional algebra is Morita equivalent with a quotient of a path algebra by an admissible ideal. Many examples including Brauer algebras will be included. 2. Representations and modules In this part we shall discuss the representations of a quiver, or of a quiver with relations. We shall see that modules over an algebra can be represented as a family of vector spaces together with a family of linear maps. In particular, projective modules and injective modules can be illustrated by diagrams in many cases, and calculation of global dimensions will be shown by examples. Gabriel’s classiflcation theorem on flnite-dimensional path algebras will be introduced. 3. Homological dimensions We shall introduce a procedure to get a pairs of algebras B and A such that B is a subalgebra of A with rad(B) = rad(A), and thenRead More
Publication Year: 2007
Publication Date: 2007-01-01
Language: en
Type: article
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