Lie Groups and Representations II

Call number: | 62723 | |

Room/Time: | MW 2:40pm--3:55pm, 407 Math | |

Instructor: | Mikhail Khovanov | |

Office: | 620 Math | |

Office Hours: | Walk-in or by appointment | |

E-mail: | khovanov@math.columbia.edu | |

TA: | Yakov Kononov, ik2402@columbia.edu | |

Webpage: | www.math.columbia.edu/~khovanov/lieGroups2018 | |

Verma modules and classification of irreducible representations of simple Lie algebras. Casimir element and complete reducibility of representations. Center of the universal enveloping algebra and the Harish-Chandra theorem.

Finite-dimensional representations of simple Lie algebras. Kostant partition function. Weyl character formula.

Characters of irreducible representations of sl(n) and Schur functions. Schur-Weyl duality between representations of sl(n) and the symmetric group. Classification of symmetric group representations. Combinatorial formulae. Jucys-Murphy elements and Young basis in irreducible representations. Symmetric functions.

Structure and decomposition of tensor products of irreducible representations. Examples for sl(2) and for rank 2 Lie algebras.

Gelfand pairs.

Clifford algebras and spin representations.

Representation theory in the non-semisimple case. Representations of Artinian algebras. Projective functors in categories of highest weight representations.

If time allows: Hopf algebras and quantum groups. Coxeter groups and Hecke algebras. Simple Lie algebras over integers. Lie algebra cohomology.

** References: **

J.Humphreys, *Introduction to Lie algebras and representation theory.*

A.Knapp, * Lie groups: Beyond an introduction.*

D.Bump, * Lie groups. *

W.Fulton and J.Harris, * Representation Theory: A First Course. *

Shlomo Sternberg,
Lie Algebras

Vera Serganova,
Representation theory: representations of finite groups, symmetric groups, GL(n),
quivers.