\documentstyle[12pt]{article} \topmargin-1cm \textheight 22cm \begin{document} \begin{center} {\bf\huge A course on Lie groups, Lie algebras and Representation Theory \\[0.2cm] (5p,C-D)} \end{center} \vspace{0.3cm} \noindent \begin{tabular}[c]{ll} {\it Lecturers}: & Sergei Silvestrov { (SS)} \\ & (e-mail: sergei@math.kth.se, tel: 08-790 6662) \\[0.1cm] & Michail Shapiro { (MSh)} \\ & (e-mail: mshapiro@math.kth.se, tel: 08-790 6693) \\[0.3cm] {\it Time}: & Spring 1997, weeks 4-21.\\ & The first lecture will take place on \\ & Friday, January 24, 1997, 15.15-17.00 \\ & in the lecture room 3721, KTH, Department of Mathematics,\\ & Lindstedtsv{\"a}gen 25, (Klocktornet).\\ & The time for other lectures will be decided at \\ & the first lecture. \end{tabular} \vspace{0.5cm} Representation theory and the theory of Lie groups and Lie algebras play an important role in modern mathematics and physics. In this course we will learn basic notions and theorems of the subject. The lectures will contain examples as well as general theory. We hope that after attending the course you will become interested in representation theory and in the theory of Lie groups and Lie algebras, and will be prepared to use actively the obtained knowledge both in relation to problems in these or other areas of mathematics and physics. The course is intended in the first place for doctoral and advanced undergraduate students. However everybody else who is interested in the subject is VERY WELCOME to join the lectures. \begin{center} {\sc Contents} \end{center} \begin{itemize} \item[I] {\it Lectures 1-4} (SS, MSh): {\bf Basic Representation Theory.} Rotation group and its representations. Differential equations invariant under rotations and their solution. Representation theory: Basic concepts methods and theorems. \item[II] {\it Lectures 5-10}: {\bf Lie algebras} \item[ 1)] {\it Lecture 5} (SS): Definition and examples of Lie algebras; Structure constants, complex and real Lie algebras; Subalgebras, ideals, center, direct sums and quotient algebras; Homomorphisms, isomorphisms, automorphisms, derivations, adjoint algebra, semidirect product of Lie algebras. \item[ 2)] {\it Lecture 6} (SS): Representations of Lie algebras; Universal enveloping algebra of a Lie algebra. Bases in Universal enveloping algebras. \mbox{Poincar{\'e} - Birkhoff - Witt} theorem. Ado's theorem. Enveloping field of a Lie algebra. \item[ 3)] {\it Lecture 7} (SS): Classification of Lie algebras up to isomorphism; Classification of all Lie algebras of $dim \leq 3$; Problem of classification of Lie algebras of $dim \geq 3$. Solvable, nilpotent, simple and semisimple Lie algebras. \item[ 4)] {\it Lecture 8} (MSh): Theorems of Lie and Engel. The Killing form. Cartan's criterion of semisimplicity. Decomposition of semisimple Lie algebras into direct sum of simple algebras. \item[ 5)] {\it Lecture 9-10} (SS, MSh): Classification of simple complex Lie algebras. Root systems and Dynkin diagrams. Highest weight representations. \item[III] {\it Lectures 11-13} (SS, MSh): {\bf Lie groups.} \item[ 1)] {\it Lecture 11}: Differentiable manifolds. Tangent spaces and vector fields. Transformation of vector fields. Lie groups and subgroups of a Lie group: definitions and examples. The structure constants. \item[ 2)] {\it Lecture 12}: The Lie algebra of a Lie group. Transformation groups and generators of one-parameter subgroups. Correspondence between Lie groups and Lie algebra. Exponential mapping. Adjoint group and adjoint algebra. Lie algebras of left- and right-invariant vector fields. \item[ 3)] {\it Lecture 13}: Invariant tensors and center of enveloping algebras of a Lie algebra. Gelfand theorem. Casimir elements. \item[IV] {\it Lectures 14-17 $^{*}$} (SS): {\bf Infinite-dimensional representations.} \item[ 1)] {\it Lecture 14:} Representations of Lie algebras and Lie groups in a Hilbert space. Representation of Lie and Enveloping algebras by unbounded operators. G{\aa}rding's theory. \item[ 2)] {\it Lecture 15:} Analytic vectors for linear operators and representations of Lie algebras and Lie groups. \item[ 3)] {\it Lecture 16:} Integrability of representations of Lie algebras. Nelson's theorem. \item[ 4)] {\it Lecture 17:} Integrability of Lie algebra representations. Flato-Simon-Snellman-Sternheimer theory ($FS^3$-theory). \end{itemize} \vspace{0.5cm} A number of additional problems-oriented seminars will be organized during the course. \newpage \begin{center} {\sc Literature}\end{center} \begin{itemize} \item[1.] A. O. Barut, R. Raczka, Theory of Group Representations and Applications, Polish Scientific Publishers, Warszawa, 1977. \item[2.] M. Boij, D. Laksov, An Introduction to Algebra and Geometry via Matrix Groups, Lecture Notes, Department of Mathematics, KTH, Stockholm, 1995. \item[3.] I. M. Gelfand, R. A. Minlos, Z. Ya. Shapiro, Representations of the Rotation Group and of the Lorentz Group and their Applications, MacMillan, New York, 1963. \item[4.] M. Goto, F. D. Grosshans, Semisimple Lie algebras, Marcel Dekker, INC., New York and Basel, 1978. \item[5.] A. A. Kirillov, Elements of the Theory of Representations, Heidelberg, Springer-Verlag, 1975. \item[6.] M. A. Najmark, Theory of Group Representations, Marcel Dekker, INC., Berlin, Heidelberg, New York, Springer-Verlag, 1982. \item[7.] J.-P. Serre, Lie algebras and Lie groups, 1964 lectures given at Harvard University, 1992. \item[8.] N. Ja. Vilenkin, Special Functions and the Theory of Group Representations, Amer. Math. Soc., Providence, 1968. \item[9.] N. Ja. Vilenkin, A. U. Klimyk, Representation of Lie Groups and Special Functions, Vol. 1, Kluwer Academic Publishers, Dordrecht, Boston, London, 1991. \item[10.] D. P. Zhelobenko, Compact Lie groups and their Representations, AMS, Transl. Math. Monogr. 40, 1973. \end{itemize} \vspace{1cm} \begin{center} !!!!!!! WELCOME !!!!!!! \end{center} \end{document}