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\begin{center}
	 { \Large \bf 
Topics for oral qualifying exam for Tom Robinson\\
\vskip .5em
 Spring, 2006} 
\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 Vertex operator construction of the affine Lie algebras
corresponding to $A_n$, $D_n$ and $E_n$.

\end{enumerate}
\goodbreak

\item The fundamentals of logarithmic tensor product theory for
generalized modules for a conformal vertex algebra.

\begin{enumerate}

\item The notions of conformal and M\"obius vertex algebra.

\item The notions of module and of generalized module.

\item Opposite vertex operators, and contragredient modules and
generalized modules.

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

\item Logarithmic formal calculus.

\item The notion of $P(z)$-tensor product of generalized modules.

\end{enumerate}

\end{enumerate}

\vskip 1.5cm


\begin{center}
{\bf Minor topic: Lie algebras}
\end{center}

\begin{enumerate}

\item Elementary notions and basic theory

\begin{enumerate}

\item Definitions, examples, representations, modules

\item Solvable, nilpotent, simple and semisimple Lie algebras and the
Killing form

\item Lie's theorem

\item Engel's theorem

\item Cartan subalgebras

\item Cartan's criteria for semisimplicity and solvability

\item Semisimple Lie algebras as direct products of simple Lie
algebras

\item Complete reducibility of modules for semisimple Lie algebras

\end{enumerate}

\item Semisimple Lie algebras and root systems

\begin{enumerate}

\item Representations of $sl(2)$

\item Root space decompostion

\item Axiomatics of root systems; simple roots; Weyl group

\item Classification

\item Construction of root systems

\end{enumerate}
\goodbreak

\item Universal enveloping algebras

\begin{enumerate}

\item Construction of the universal enveloping algebra

\item The Poincar\'e-Birkhoff-Witt theorem

\item Free Lie algebras

\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[HLZ]{HLZ} Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic
tensor product theory for generalized modules for a conformal vertex
algebra, to appear.

\bibitem[H]{H} J. Humphreys, Introduction to Lie Algebras and
Representation Theory, Second Printing, Revised, Springer-Verlag,
1972.

\bibitem[LL]{LL} J. Lepowsky and H. Li, Introduction to Vertex
Operator Algebras and Their Representations, Progress in Math.,
Vol. 227, Birkh\"auser, Boston, 2003.

\bibitem[LM]{LM} J. Lepowsky and G. McCollum, Elementary Lie algebra
theory, Yale Univ. Lecture Note Series, 1974, 138 pgs.; corrected,
1982, Rutgers Univ.

\end{thebibliography}

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