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\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}
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