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{\bf {\large{Oral Qualifying Exam Syllabus\\[2mm]}}}
Debajyoti Nandi\\[2mm]
November 30, 2006\\[2mm]
Committee: Profs. R. Wilson, J. Lepowsky, V. Retakh, E. Taft\\[1cm]
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{\bf Major Topic: Lie Algebras and Their Representations}

\begin{enumerate}

\item Classification of semisimple Lie algebras over $\mathbb{C}$
\begin{itemize}
	\item Solvability, nilpotency (Engel's theorem, Lie's theorem, etc.)
	\item Structure of semisimple Lie algebras
	\item Representation of $\mathfrak{sl}(2,\mathbb{C})$
	\item Root system and root space decompostion
	\item Classification of semisimple Lie algebras
\end{itemize}

\item Existence of classical Lie algebras

\item Representation theory of semisimple Lie algebras
\begin{itemize}
	\item Universal enveloping algebra and PBW theorem
	\item Weights, standard cyclic modules, finite dimensional modules
	\item Character formulas
\end{itemize}

\item Kac-Moody algebras
\begin{itemize}
	\item Constructions
	\item Representation Theory
\end{itemize}

\item Construction of affine algebras and extensions of $\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]$

\item Construction of Basic Modules.
\begin{itemize}
	\item Lie Algebra Level
	\item Vertex Algebra Construction
\end{itemize}



\end{enumerate}


~\\[1cm]
{\bf Minor Topic: Hopf Algebras}
\begin{itemize}
\item Definition of Hopf algebra (comodule, bimodule, antipode)
\item Coideals and comodules
\item Duality -- $A^0$, $C^*$
\item Definition of integrals and smash product
\item $H$-module algebra and coalgebra
\item Fundamental theorem of coalgebra
\item Coradical filtration
\item Quantum groups: $M_q(2)$, $M_q(n)$, $det_q(X)$, $GL_q(2)$, $SL_q(2)$
\end{itemize}

\pagebreak

{\bf References}

\begin{enumerate}
\item Frenkel, I., Lepowsky, J., Meurman, A. {\em Vertex Operator Algebras and the Monster}.
Pure and Applied Math., Vol. 134, Academic Press, Boston, 1988.

\item Humphreys, James E.  {\em Introduction to
Lie Algebras and Representation Theory}.  Springer-Verlag, New York,
1972.

\item Kac, V. G. {\em Infinite Dimensional Lie Algebras}, 3rd ed.,
Cambridge University Press, Cambridge, 1990.

\item Lepowsky, J. {\em Lectures on Kac-Moody Lie algebras}. Universit\'{e} Paris VI, Spring, 1978.

\item Lepowsky, J., Wilson, R.  {\em Construction of the Affine Lie Algebra $A_{1}^{(1)}$}.
Commun. math. Phys. 62, 43--53(1978).

\item Montgomery, S. {\em Hopf Algebras and their Actions on Rings}. 
CBMS Reg. Conf. Series 82, Providence, R.I., 1993.

\item Sweedler, M. E. {\em Hopf Algebras}. Benjamin New York, 1969.

\end{enumerate}


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