Finite dimensional algebras and linear representations

elective courses

The lecture aims to present the classical results concerning the structure and linear representation theory of

finite dimensional algebras over fields. The following will be discussed: correspondence between theory of

modules and representation theory, simple modules, radical algebras and classification of semi-simple associative algebras.

Applications will be given to the representation theory of finite groups, through results concerning

group algebras and the theory of group characters. Examples of applications will be discussed. Basic information on finite dimensional Lie algebras and their representations will be given. As a tool in this theory,

universal enveloping algebras and their properties will be discussed

1. Finite dimensional associative algebras over a field.Concept and examples of algebras, finite dimensional algebras. Simple algebras and division algebras.Modules over associative algebras, semi-simple and simple modules. Radical of an associative algebra. Wedderburn’s theorem on the structure of semisimple algebras. Schur's lemma. Structure of finitely generated modules over semisimple algebras. Group algebras. Maschke theorem. Irreducible modules,Fitting’s lemma and Krull-Schmidt theorem.

2. Representations of groups

Irreducible and completely reducible representations. Traces of endomorphisms and characters. Orthogonality of characters. Integral extensions. Representations of finite abelian groups and symmetric groups. Examples of application, e.g. proof of the solvability of groups of order p^kq^n

3. Finite dimensional Lie algebras and their representations.

Definition and examples. Solvable radical. Semisimple algebras and information about a structural theorem for simple algebras over the field of complex numbers. Linear representations. Enveloping algebras and the Poincare-Birkhoff-Witt theorem. Free associative algebra and the "diamond lemma" as a tool in proof.

1. J. Browkin, Teoria Reprezentacji Grup Skonczonych, PWN Warszawa, 2010.

2. C.W. Curtis, I. Reiner Representation Theory of Finite Groups and Associative Algebras, Interscience

Publ. 1962.

3. K. Erdmann, M.J. Wildon, Introduction to Lie Algebras, Springer, 2006.

4. J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, 1980.

5. Y.T. Lam, A First Course in Noncommutative Rings , Springer-Verlag, 1991.

6. Y.T. Lam, Exercises in Classical Ring Theory, Springer-Verlag, 2003.

7. R.S. Pierce, Associative Algebras, Springer-Verlag, 1982.

8. J.-P. Serre, Reprezentacje Liniowe grup skonczonych. PWN, Warszawa 1998.

1. Knows the concepts of algebra, ideal, module and submodule over an algebra, as well as the basic constructions of algebras and modules. He knows the concept of simple and semisimple modules and their characterizations. He can describe elements of ideals and submodules generated by sets and give various examples of algebras.

2. Knows the concept of homomorphism, algebras and modules, isomorphism’s theorems and imbedding of algebras in matrix algebras and the Schur’s lemma.

3. Knows the notion of a radical of an algebra and of semisimple algebra, theorems of Wedderburn and Maschke. Can describe the structure of the finite dimensional modules over finite dimensional algebras semisimple algebras. Can use these concepts and facts to describe the structure of finite dimensional algebras and classification of low-dimensional algebras;

4. Knows the concept of an irreducible module, the notion of local algebra and the relationship between these concepts. He knows the Krull-Schmidt theorem.

5. Knows the concept a representation of finite groups and finite dimensional algebras, irreducible representations and completely reducible, regular representation, and characters of representations. Can express the concept of a group representation in the language of modules over the group algebra of that group. Knows the theorem on the orthogonality of irreducible characters of complex representations of finite groups and the theorem that complex representations of a finite group, having equal characters, are equivalent. He knows the solvability theorem of groups whose orders are products of powers two prime numbers;

6. Knows the basic theorems concerning representations of finite groups over the field of complex numbers and the relations of their degrees and the number of not equivalent representations with the appropriate parameters of groups and decomposition of group algebras over the field of complex numbers to the direct product of matrix algebras . Can use these theorems to describe group algebras of groups of low order;

7. Knows the concept of a finite dimensional Lie algebra and basic examples of such algebras. Can describe low-dimensional complex Lie. He knows the concept of the radical a Lie algebra and the concepts of simple and semi-simple algebras and can give examples of such algebras. He knows the concept of the Killing forms and can apply them to the study of semisimple, finite dimensional Lie algebras over the field of complex numbers. He knows the concept of a representation of a Lie algebra. He knows the concept of the universal enveloping and its basic properties, including the Poincare-Birkhoff-Witt theorem.