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\center{Oral Qualifying Exam Syllabus} \center{Colleen Duffy}
\center{28 February 2006} \center{Committee: Profs. R. Wilson, R.
Goodman, V. Retakh, E. Taft}

\bigskip

Major Topic: Noncommutative Algebra

\begin{enumerate}

\item General noncommutative ring theory ([Lam] \S\S 1-4, 7-9)

\begin{itemize}

\item Semi-simple modules and rings
\item Wedderburn-Artin Theory
\item Jacobson Radical
\item modules over kG/representations of finite groups, characters
\item linear groups: Burnside's Theorem
\end{itemize}

\item Quasideterminants ([GGRW])

\begin{itemize}

\item definition - in terms of inverses and recursive
\item properties - e.g. row/column relations, Sylvester's theorem
\item applications - e.g. Vandermonde quasideterminant, Vieta theorem, symmetric
functions, quasi-Plucker coordinates

\end{itemize}

\item Algebras related to roots of equations

\begin{itemize}

\item $Q_n$, $A(\Gamma)$ - definition, describe linear basis[GRSW]
\item Bergman’s Diamond Lemma ([Bergman] \S\S 1-3)
\item factorization of twisted polynomial rings: remainder and product theorems,
definition and example of Wedderburn polynomials ([LL] \S\S 1-2;
[LL2] \S\S 1-3)
\item Koszul algebras: definition, dual of, Hilbert series of ([Froberg] \S\S
1-2)

\end{itemize}

\item Lie Algebras ([GW] Ch. 1-2) ([Humphreys]\S\S1-8)

\begin{itemize}

\item Lie group and Lie algebra correspondence
\item classification of finite dimensional semisimple algebras over the complex numbers
\item classification of irreducible representations by highest weight (isomorphism)
\item some explicit examples
\item PBW Theorem

\end{itemize}

\end{enumerate}

Minor Topic: Hopf Algebras

\begin{itemize}

\item Definition of Hopf algebra (coalgebra, bialgebra, 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 Coalgebras

\end{itemize}

\pagebreak

\center{References}

\begin{enumerate}

\item [Bergman] Bergman, George M.  "The Diamond Lemma for Ring Theory."  Advances in Mathematics, 29 (1978) 178-195.

\item [Froberg] Froberg, R.  "Koszul Algebras." 1998.

\item [GGRW] Gelfand, Israel, Sergei Gelfand, Vladimir Retakh, and Robert Wilson.  "Quasideterminants."  Advances in Mathematics 193
(2005) 56-141.

\item [GRSW] Gelfand, Israel, Vladimir Retakh, Shirlei Serconek, and Robert Wilson.  "On a Class of Algebras Associated to Directed Graphs."


\item [GW]  Goodman, Roe and Nolan Wallach.  \underline{Representations and Invariants of the Classical Groups}.
Encyclopedia of Mathematics and its Applications, Cambridge, 1998.

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

\item [Lam] Lam, T.Y.  \underline{A First Course in Noncommutative Rings}.  2nd ed. Springer-Verlag, New York, 2001.

\item [LL] Lam, T.Y. and A. Leroy.  "Vandermonde and Wronskian Matrices over Division Rings."  Journal of Algebra 119, 308-336 (1988).

\item [LL2] Lam, T.Y. and Andre Leroy.  "Wedderburn Polynomials over Division Rings, I."  Contemporary Mathematics.

\end{enumerate}

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