Lie Groups, Lie Algebras

and Representations

Instructor:Guanghua JI

**Aims:**Lie theory is currently a very active area of research and provides many interesting examples and patterns to other branches of mathematics. To introduce students to fundamental results of Lie theory(Lie correspondence thm, Cartan-Weyl theory and basic representations theory) as an essential part of general mathematical culture and as a basis for further study of more advanced mathematics.

Textbook:W. Rossmann, Lie Groups: An introduction through linear groups (Knapp's review,Trapa's review)

**Classroom and Time:**B124, B108, ZhiXin Building; Mon 7-8, Thu 3-4

**Prerequisites:**Calculus, linear algebra, group theory

**Grading:**The course grade will be based homework(50%), final exam(30%) and final project(20%)

References:

1, K. Tapp, Matrix Groups for Undergraduates, AMS, 2005.

2, A. Baker, Matrix groups, an introduction to Lie group theory, Springer.

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

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

5, A. W. Knapp, Representation Theory of Semisimple Groups: an Overview Based on Examples, 2001.

6, A. W. Knapp, Lie Groups, Beyond an Introduction, Birkhauser, 2002.

Syllabus:

**Chapter I. The Exponential Map**

2/28,Prerequisties and background:Linear algebra and analysis review.

3/4,(1.1), The exponential functionand the logarithm functions: (1), definitions, properties; (2), computing the exponential of a matrix: diagonalizable, nilpotent, arbitrary.

3/7, (3), Lie product formula; (4), det(exp(X))=exp(trX).(1.2),One-parameter subgroup:definition, A(t) is one parameter subgroups iff A(t)=exp(tX) for some X with X=A'(0).

3/11,(1.3), The adjoint representation:adjoint repe, Lie bracket, derivative of the exp mapping, eigenvalues of adX.

3/14, (1.4),The Baker-Campbell-Hausdorff formula:integral form, series form, remainder term and applications.

**Chapter II, Linear Lie Groups**

3/18,(2.1), Linear lie groups: definitions and examples: GL(n,k), SL(n,k), O(n), U(n), SO(n), SU(n). Homework: p42.

3/21,(2.2), The linear lie algebra:(1), The Lie algebra of a linear groups:Definitions: linear lie algebra is a real vector space, closed under bracket operation, dimensional of linear groups, examples. Homework: p52. 1,3.

3/25, (2), Abstract lie algebra, Lie subalgebra, simple Lie algebra, Aut(g), Derviation. Homework:p52, 4,5,6,7,8,9.

3/28,(2.3), Linear groups are manifolds:(1), Lie alge is the tangent space: The exp maps g into G; g={X|exp(tX)\in G, for all t\in R}=TG. (2), Manifolds of R^n, definition. Homework: p57

4/1, (3), linear groups are manifolds: exp is a local diffeomorphism at 0 from g to G; (4), Canonical Coordinates: the first kind, second, third.(2.4), The topology of linear groups:(1), Topological groups: definitions and examples GL(n, R); (2), Compactness: bounded and closed, O(n), SO(n), U(n), SU(n). Homework: p65

4/8, (3), Connectedness: connected, path-connected, any connected linear lie gp is generated by any nbd of the identity(in particuar by exp(g)), the identity component is a clopen normal and unique open connected subgp of G; examples: GL(n, C), SL(n, k), SO(n), U(n). (4), Simply connected: Definition, Example: SU(n).

4/11,(2.5), Homomorphisms:(1) Differentials of Homomorphisms: continuous is smooth, from lie gp homo to lie alg homo; Hom(G, H) -- Hom(g, h); two isomorphic linear lie groups have isomorphic lie algebras.

4/15, (2), Locally mappings: f is locally bijective iff df is bijective; (3), Example: The adjoint representations. Ad: G -- Aut(g), dAd=ad.

4/18, (2.6),The Lie correspondence theorem:(1), Subgroups and subalgebras: Baire's covering lemma, Lie correspondence theorem, connected linear groups to linear lie algebras. (2), Connected abelian lie groups: lattices, discrete subgroup, structure of connected abelian lie groups.

4/22,(2.7),Covering theory of Lie groups:(1), Covering groups: definition, properties. (2), Lie's second theorem: homo of simply connected lie gps and homo of their lie algebras; If G is simply-connected, then dF: Aut(G) --- Aut (g) is an isomorphism of lie groups.

**Chapter III, Abstract Lie Algebras**

4/25,(3.1), Ideals, Centralizer and Normalizers: (1), Ideals: examples, quotient algebra, isomorphism theorem, connected normal subgroup iff ideal, commutator subalgebra; (2), Centralizer and normalizer: definitions and properties.

4/28,(3.2), Solvable Lie algebras: (1) Sovable Lie algebra: subalgebra, image of homo of lie algebras, radical. solvable lie groups; (2), Lie's theorem: exists a simultaneous eigenvetor.

5/2,(3.3), Nilpotent Lie algebras: (1) Nilpotent Lie algebra: subalgebra, image of homo of lie algebras, nilpotent lie groups; (2), Engle's theorem: g nilpotent iff adX nilpotent.

5/6, (3.4),Semisimple Lie algebras: (1) Definitions and Examples: simple, semisimple: has no zero abelian (solvable, or nilpotent) ideals. g/radg is semisimple. (2) Killing forms: def and properties. (3), Cartan's criterion: if Tr(XY)=0, then g is solv; the killing form vanishes, then g is solv; semisimple iff nondegenerate. (4), Reducitive lie algebra.

5/9,(3.5), Compact Lie algebras:(1) Defintion: Killing form<0 with trivial center. (2), cartan subalgebra

**Chapter VI, Representation theory of Lie groups and Lie algebras**

5/13,(4.1), Haar measure on locally compact groups: (1), Haar measure and Haar integral: existence and uniqueness. (2), Modular function: compact gp is unimodular, GL(n,R). (3), Haar measure on a linear lie group.

5/16,(4.2), Representations: definitions and examples(1), Baisc Defintions: invariant space, subrepr, quotient repr, irreducible, reducible, unitary, direct sum, completely reducible, intertwining map, equivalent. (2), Contragredient representations. (3), Tensor product of reprs. (4), Compact group: is unitary rep and completely reducible. (4), Examples: Left and right regular rep of compact G.

5/20, (4.3), Peter-Weyl Theorem(1), Schur's lemma. (2), An irr repr of a compact gp is finite-dim. (3), Schur's orthogonality relations: matrix coefficients. (4), Peter-Weyl Theorem: matrix coefficients form a countable complete set of ortho functions.

5/23, (5),Linearity of compact lie groups: Gelfand-Raikov theorem, every comapct lie group admits a faithful representation. (6), Plancherel's formula for a compact group.

5/27,(4.4), Characters and class functions(1), Characters: Shur's orthogonality relations for characters, irreducibility criterion. (2) Class functions: Peter-Weyl thm for class functions, Fourier expansion of class functions. (3), Isotypic component. (4), Burnside's Theorem.

5/30,(4.5), Maximal Torus, Weights and Roots(1), Maximal torus: Cartan theorem. (2),Weights and Roots: Weight(roots) spaces decomposition.

6/3,(4.6), Irreducible rep of the unitary group U(n): Weyl's Formulas(1), Weyl's integration formula (2), Highest weight theorem. (3), Weyl's character formula, (4), Weyl's dimension formula.

6/6,(4.7), The Representations of Lie algebras(1), Weyl unitarian trick. (2), Complete reducibility

6/13, (4.8), The irreducible representations of SU(2), SO(3) and SL(2,R). Examples

final exam

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