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\begin{center}
	 { \Large \bf 
Topics for oral qualifying exam for Corina Calinescu\\
\vskip .5em
 Spring, 2003} 
\end{center}

\begin{center}
{\bf Major topic: Vertex operator algebras}
\end{center}

\begin{enumerate}

\item Definitions and properties.

\begin{enumerate}

\item Formal calculus.

\item The notions of vertex algebra and of vertex operator algebra,
and basic properties.

\item Rationality, commutativity and associativity; equivalence 
of various formulations,
including ``weak'' formulations.

\end{enumerate}


\item Representations of vertex (operator) algebras.

\begin{enumerate}

\item The notion of module and basic properties.

\item Weak vertex operators.

\item The structure of the canonical weak vertex algebra. Local
subalgebras and vertex subalgebras of the canonical weak vertex algebra. 

\item The equivalence between modules and representations.

\item General construction theorems for vertex (operator) algebras 
and modules.

\end {enumerate}

\item Examples of vertex (operator) algebras and modules.

\begin{enumerate}

\item Vertex (operator) algebras and modules based on
the Virasoro algebra.

\item Vertex (operator) algebras and modules based on
affine Lie algebras.

\item Vertex (operator) algebras and modules based 
on Heisenberg Lie algebras.

\item Vertex (operator) algebras and modules on
even lattices.

\item Classification of the irreducible modules for the simple
vertex operator algebras associated to standard modules.

\end{enumerate}

\item The notions of intertwining operator and fusion rule,
basic properties.

\end{enumerate}

\vskip 2cm


\begin{center}
{\bf Minor topic: Kac-Moody algebras}
\end{center}

\begin{enumerate}

\item Definitions and properties.

\begin{enumerate}

\item Triangular and root space decompositions.

\item The invariant bilinear form and the generalized
Casimir element.

\item Integrable representations of Kac-Moody algebras
and the Weyl group.

\end{enumerate}

\item A classification of generalized Cartan matrices.

\item Real and imaginary roots. 

\begin{enumerate}

\item Definitions and properties.

\item An explicit description of the root system of a Kac-Moody algebra.

\end{enumerate}

\item Affine algebras.

\begin{enumerate} 

\item The normalized invariant form, the root system, and the Weyl group.

\item Affine algebras as central extensions of loop algebras.

\end{enumerate}

\item The representation theory of Kac-Moody algebras.

\begin{enumerate}

\item The category $ \it{O}$. Highest-weight modules and Verma modules.

\item The irreducibility and complete reducibility in $ \it{O}$.

\item Formal characters of modules from $ \it{O}$. 

\item Integrable highest-weight modules over Kac-Moody algebras. 
The character formula, the denominator formula and the multiplicity
formula.

\item The denominator formula for $ \widehat{sl_2(C)}$. 
The Jacobi triple product identity.

\item On the defining relations for a symmetrizable Kac-Moody algebra. 

\end{enumerate}

\end{enumerate}


\begin{thebibliography}{FLM2}

\bibitem[FHL]{FHL}
I. Frenkel, Y.-Z. Huang and J. Lepowsky, On Axiomatic Approaches
to Vertex Operator Algebras and Modules, Memoirs Amer. Math. Soc. 104
(1993).

\bibitem[FLM]{FLM} 
I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras
and the Monster, Academic Press, 1988.

\bibitem[K]{K}
V. Kac, Infinite dimensional Lie algebras, third edition,Cambridge
University Press, 1990.

\bibitem[LL]{LL}
J. Lepowsky and H. Li, Introduction to vertex operator algebras 
and their representations, to appear.

\bibitem[L]{L}
H. Li, Local systems of vertex operators, vertex superalgebras and
modules, J. Pure and Applied Algebra 109 (1996), 143-195.


\end{thebibliography}

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