Basic objects and notions of representation theory: Associative algebras. Algebras definedby generators and relations. Group algebras. Quivers and path algebras. Lie algebras and enveloping algebras. Representations. Irreducible and indecomposable representations. Schur’s lemma. Representations of sl(2).Basic general results of representation theory. The density theorem. Representations of finite dimensional algebras. Semisimple algebras. Characters of representations. Jordan- Holder and Krull-Schmidt theorems. Extensions of representations.Representations of finite groups, basic results. Maschke’s theorem. Sum of squares formula. Duals and tensor products of representations. Orthogonality of characters. Orthogonality of matrix elements. Character tables, examples. Unitary representations. Computation of tensor product and restriction multiplicities from character tables. Applications of representation theory of finite groups.Representations of finite groups, further results: Frobenius-Schur indicator. Frobenius determinant. Algebraic integers and Frobenius divisibility theorem. Applications to the theory of finite groups: Burnside’s theorem. Induced representations and their characters (Mackey formula). Frobenius reciprocity. Representations of GL(2; Fq). Representations of the symmetric group and the general linear group. Schur-Weyl duality. The fundamentaltheorem of invariant theory.Representations of quivers. Indecomposable representations of quivers of type A1, A2, A3,D4. The triple of subspaces problem. Gabriel’s theorem. Proof of Gabriel’s theorem: Simply laced root systems, reflection functors.
