**Instructor:** Professor Victor Kac

**Email:** kac [at] math . mit .edu

**Office:** Room 2-176

**Office Hours:** By Appointment

**Grader:** Gurbir Dhillon gdhillon@mit.edu

**Lectures:** TR 2:30-4:00 2-143

## COURSE OVERVIEW

Lecture 1, 2: | Introduction, basic definitions |

Lecture 3-5: | Engel and Lie theorems |

Lecture 6-9: | Cartan subalgebras and Chevally's conjugacy theorem |

Lecture 10: | Cartan's criterion |

Lecture 11-14: | Structure of semi-simple Lie algebras |

Lecture 15-20: | Classification and construction of simple Lie algebras |

Lecture 21: | Compact form and Weyl group |

Lecture 22: | Universal enveloping algebra, PBW theorem, Casimir element and cohomology of Lie algebras |

Lecture 23: | Weyl, Levi and Maltsev theorems |

Lecture 24: | Verma modules |

Lecture 25, 26: | Finite-dimensional representations of semisimple Lie algebras. Weyl character and dimension formulae |

## RECOMMENDED BOOKS

- J.P. Serre, "Complex Semi Simple Lie Algebras"
- J.E. Humphreys, "Introduction to Lie Algebras and Representation Theory"
- A.L. Onishchik, E.B. Vinberg, "Lie Groups and Algebraic Groups"

## COURSE REQUIREMENTS

- Typing two lectures ( using TeX is preferred, MS Word is ok; contact typers of previous lectures for templates). Lecture notes should be handed in within a week of the date of the lecture.
- Doing exercises given in the course of lecture; collected and graded weekly. The exercises from the previous two lectures are due every Tuesday (if holiday, then the next Tuesday).
- No final exam