**Lie Groups and Representations**

**Textbook:** M. R. Sepanski, **Compact Lie Groups**, GTM 235, Springer.

**Classroom and Time: **C501, ZhiXin Building; Fri. 2-4 (9:00 - 12:00).

**Prerequisites: **Group theory, Basic topology, functional analysis

**Grading: **The course grade will be based homework(50%) and final exam(50%).

**References:**

1, R. Goodman and N. R. Wallach, Symmetry, Representations, and Invariants, GTM 255, Springer, 2009..

2, B. Hall, Lie Groups, Lie Algebras, and Representations, GTM 222, 2004.

3, J. Faraut, Analysis on Lie Groups, an introduction, CSAM 110, Cambridge University, 2008.

4, W. Rossmann, Lie Groups: An introduction through linear groups, Oxford Univ Press, 2002.

**Syllabus:**

**Chapter I. Compact Lie Groups**

3/06, 1, Topology Groups. 2, Connectedness. 3, Manifolds. 4, Lie Groups. HW: 4, 5, 16, 17.

Chapter II, Representations

3/13, 1, Definitions. 2, Schur's Lemma. HW: 2, 15(a), 18.

3/20, 3, Unitary representations. 4, Some reprentations of SU(2). HW: 1.43; 2.26.

**Chapter III, Harmonic Analysis**

3/27, 1, Schur orthogonality relation. 2, Characters. HW: 1, 2, 3, 9.

4/03, 3, Infinite dimensional representations: irr unitary repr of a compact group is finite dim. HW: 13, 15,18, 19.

4/10, 4, The Peter-Weyl Theorem: Convolution; P-W Thm; Faithful repr; Class functions; Irr repr of SU(2). HW

**Chapter IV, Lie Algebras of Linear Lie Groups**

4/17, 1, Lie algebras of Linear Lie groups: The exponential mapping; The logarithm mapping; Lie algebras and examples. HW

4/24, 2, The Lie coresspondence: Lie algebra homomorphisms. HW

5/08, 3, The Baker-Campbell-Hausdorff formula, subalgebras and subgroups. HW

**Chapter V, Abelian Lie Subgroups and Structure**

5/15, Maximal Tori and Cartan subalgebras. HW

5/22, Lie Algebra structure. HW

Chapter VI, Roots and Associated Structures

5/29, Root Theory, repr of Lie algebra

6/05, The standard sl(2, C) triple

6/12, Weyl groups

**Chapter VII, Highest Weight Theory**

6/19, Weyl integration formula

6/26, Weyl character formula