00006 - Advanced Algebra

Course contents

1. REPRESENTATION THEORY OF FINITE GROUPS

• Representations of a group G, G-modules and G-submodules.
• Group algebra, permutation representation, regular representation.
• Irreducibles submodules, complete reducibility. Maschke's theorem, G-homomorphisms and Schur's lemma.
• Structure of the centralizing algebra of a matricial representation.
• Tensor product of vector spaces. Tensor product of matrices.
• Characters and class functions. Character tables. Orthogonality relations.
• Structure of the group algebra.
• Character products. Character and normal subgroups.
• Algebraic integers.
• Representations of symmetric groups.
  

2. LIE ALGEBRAS

• Lie algebras. Linear Lie algebras. Classical, triangular and diagonal Lie algebras.
• Derivations and adjoint representation. Abstract Lie algebras.
• Ideals, center, derived algebra. Simple Lie algebra. Normalizer and centralizer. Solvable Lie algebras,
• Nilpotent Lie algebras. Engel's theorem.
• Lie's theorem. Jordan decomposition.
• Cartan's criterion. Killing form.
• Decomposition of a semisimple Lie algebra. Abstract Jordan decomposition.
• Representations and irreducibility. Schur's lemma.
• Casimir element. Weyl's theorem.
• Representations of sl_2.
• Toral subalgebras. Cartan decomposition of a semisimple Lie algebra.
• Orthogonality, integrality and rationality properties.
• Mentions on the classification
  

Readings/Bibliography

1. BRUCE E. SAGAN. The Symmetric group. Representations, Combinatorial algorithms, and Symmetric functions. Second edition. Graduate Texts in Mathematics, 203. Springer-Verlag, New York,
   2001.
2. JAMES E. HUMPHREYS. Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics, 9. Springer-Verlag, New York, 1972.

Assessment methods

The exam consists in the resolution of two compulsary assignments given during the course and an oral exam.

Links to further information

http://www.dm.unibo.it/~mmorigi/algsup_m/algsup09-10.html

Office hours

See the website of Fabrizio Caselli

See the website of Marta Morigi