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\par\noindent Proposed topics for the Oral Qualifying Exam
\par\noindent Rutgers University, Department of Mathematics
\par\noindent Candidate: Aaron Lauve
\par\noindent Committee: Earl Taft, Robert Wilson, Vladimir Retakh
\par\noindent {\bf Date: January 29, 2002}
\vskip 2cm


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\begin{center}
{\Large\emph{major topic: ``Noncommutative Rings''}}
\end{center}

\section{Fundamental Ring Theory}
\subsection{Semisimplicity}
\noindent{\bf Semisimple rings:} definitions and
examples; homological characterizations; Wedderburn-Artin theorem
(includ. Rieffel's approach); simple rings (artinian and nonartinian).  (\cite{Lam:1} \S\S1--3)\smallskip
\par\noindent{\bf J-semisimplicity:} characterizations
of the Jacobson radical; examples; artinian $J$-semisimple rings;
Hopkins-Levitzki theorem; Nakayama's lemma; von Neuman regular rings as nearly semisimple rings. (\cite{Lam:1} \S4)

\subsection{Skew fields}
\noindent{\bf Basic structure theory:} Wedderburn's finite
field theorem; commutators as generators;
Cartan-Brauer-Hua theorem; ``projective lines'' and the finite coset
question; Herstein's theorem on conjugates. (\cite{Lam:1} \S13)\smallskip
\par\noindent{\bf Classical constructions:} the real quaternions;
Hilbert's twisted Laurent series; Dickson's cyclic algebras; the
Mal'cev-Neumann construction; $U(\mathfrak g)$ and it's field of
fractions. \newline(\cite{Lam:1} \S14; \cite{Jac:1} ch. 5)\smallskip
\par\noindent{\bf More structure theory:} classification of locally
compact skew fields; structure of $p$-fields. (\cite{Wei:1} pp. 1--23)

\subsection{Free skew fields}
As fields of fractions; their universal property; embeddability in
larger free skew fields; their center. (\cite{Coh:2} pp. 224,225, 235--238)


\section{Polynomials}
\subsection{Symmetric polynomials}
The various bases; Jacobi-Trudi identity; Kostka numbers and the
Littlewood-Richardson rule; orthonormality of the Schur
polynomials. (\cite{FulHar:1} app. A)

\subsection{Roots in skew fields}
\noindent{\bf General results:} the Euclidean algorithm for skew
fields; possibility of infinite number of roots; the Gordon-Motzkin
theorem on conjugacy classes.\smallskip
\par\noindent{\bf Minimal polynomials:} Dickson's
theorem on conjugates; Wedderburn's theorem on complete splitting; criteria for infinite number of roots; the Gelfand-Retakh contribution.
(\cite{Lam:1} \S16; \cite{GelRet:2} \S3)


\section{Generators and Relations}
\subsection{Lie algebras}
\noindent{\bf Root systems:} reflections in Euclidean space; root
systems and simple roots; the Weyl group and Weyl chambers; the
Cartan matrix; $\mathfrak h^*$. (\cite{Ser:1} ch. 5)\smallskip
\par\noindent{\bf Free algebra techniques:} Serre's construction of
semisimple Lie algebras; the Elimination theorem; the
Poincar\'e-Birkhoff-Witt theorem. (\cite{Hum:1} \S18; \cite{Reu:1}
ch. 0; \cite{Jac:1} ch. 5)

\subsection{Associative algebras}
\noindent Inversion height in the free skew field; Bergman's diamond
lemma. (\cite{Reu:2} pp. 93--109; \cite{Ber:1} \S\S1--5)
\vskip 3cm

\begin{center}
{\Large\emph{minor topic: Hopf Algebras}}
\end{center}
\setcounter{section}{0}
\section{Hopf Algebras}
\noindent{\bf Basic theory:} coalgebra theory (e.g. fundamental
theorem of coalgebras); definition of Hopf algebras and
Hopf ideals; the antipode; duality ($C^*$ and $A^\circ$); density criteria. (\cite{Swe:1} \S\S1.0--4.3, 6.0, 6.1)\smallskip
\par\noindent{\bf Examples:} the group algebra and its graded dual;
$U(\mathfrak g)$; the shuffle algebra; Taft algebras. (\cite{DNR:1}
pp. 158--166;  \cite{Swe:1} pp. 247--249)\smallskip
\par\noindent{\bf Structure theory:} irreducible, simple, and pointed
coalgebras; the wedge, coradical, and coradical filtration;
Milnor-Moore (the Lie algebra/Hopf algebra correspondence). (\cite{Swe:1} \S\S8.0, 9.0, 13.0)

\section{Quantum Groups}
\noindent{\bf Quantum plane:} the quantum determinant; $\hbox{SL}_q(2)$
\& $\hbox{GL}_q(2)$; the quantum plane as comodule. (\cite{Kas:1} ch. 4)\smallskip
\par\noindent{\bf Enveloping algebras:} definition of $U_q(\mathfrak g)$; Hopf
algebra structure; triangular decomposition; representations (in
analogy to those of $U(\mathfrak g)$; complete reducibility; examples). (\cite{Jan:1} ch. 4--5A)


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%  My Orals Syllabus Bibliography File
%  Aaron Lauve
%

@article {Ber:1,
    AUTHOR = {Bergman, George M.},
     TITLE = {The Diamond Lemma for Ring Theory},
   JOURNAL = {Adv. in Math.},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {29},
      YEAR = {1978},
    NUMBER = {2},
     PAGES = {178--218},
      ISSN = {0001-8708}
}


@book {ChaPre:1,
    AUTHOR = {Chari, Vyjayanthi and Pressley, Andrew},
     TITLE = {A Guide to Quantum Groups},
      NOTE = {Corrected reprint of the 1994 original},
 PUBLISHER = {Cambridge University Press},
   ADDRESS = {Cambridge},
      YEAR = {1995},
     PAGES = {xvi+651},
      ISBN = {0-521-55884-0}
}


@book {Coh:1,
    AUTHOR = {Cohn, Paul Moritz},
     TITLE = {Skew field constructions},
      YEAR = {1977},
   series  = {London Mathematical Society Lecture Note Series},
    number = 27,
 PUBLISHER = {Cambridge University Press},
   ADDRESS = {Cambridge},
     PAGES = {xii+253}
}


@book {Coh:2,
    AUTHOR = {Cohn, P. M.},
     TITLE = {Skew Fields},
      NOTE = {Theory of General Division Rings},
 PUBLISHER = {Cambridge University Press},
   ADDRESS = {Cambridge},
      YEAR = {1995},
     PAGES = {xvi+500},
      ISBN = {0-521-43217-0}
}


@book {DNR:1,
    AUTHOR = {Dascalescu, Sorin and Nastasescu, Constantin and Raianu, Serban},
     TITLE = {Hopf Algebras},
      NOTE = {An Introduction},
 PUBLISHER = {Marcel Dekker, Inc.},
   ADDRESS = {New York, Basel},
      YEAR = {2001},
    series = {Pure and Applied Mathematics},
    volume = 235,
      ISBN = {0-8247-0481-9}
}


@book
{FulHar:1,
    AUTHOR = {Fulton, William and Harris, Joe},
     TITLE = {Representation Theory},
      NOTE = {A First Course,
              Readings in Mathematics},
 PUBLISHER = {Springer-Verlag},
   ADDRESS = {New York},
      YEAR = {1991},
    series = {Graduate Texts in Mathematics},
    volume = 129,
      ISBN = {0-387-97495-4}
}


 @incollection {GelRet:1,
    AUTHOR = {Gelfand, Israel and Retakh, Vladimir},
     TITLE = {Noncommutative {V}ieta Theorem and Symmetric Functions},
 BOOKTITLE = {The Gelfand Mathematical Seminars, 1993--1995},
     PAGES = {93--100},
 PUBLISHER = {Birkh{\"a}user Boston},
   ADDRESS = {Boston, MA},
      YEAR = {1996}
}


@article {GelRet:2,
    AUTHOR = {Gelfand, I. and Retakh, V.},
     TITLE = {Quasideterminants, {I}},
   JOURNAL = {Selecta Math. (N.S.)},
  FJOURNAL = {Selecta Mathematica. New Series},
    VOLUME = {3},
      YEAR = {1997},
    NUMBER = {4},
     PAGES = {517--546},
      ISSN = {1022-1824}
}


@article {GKLLRT:1,
    AUTHOR = {Gelfand, Israel M. and Krob, Daniel and Lascoux, Alain and
              Leclerc, Bernard and Retakh, Vladimir S. and Thibon,
              Jean-Yves},
     TITLE = {Noncommutative Symmetric Functions},
   JOURNAL = {Adv. Math.},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {112},
      YEAR = {1995},
    NUMBER = {2},
     PAGES = {218--348},
      ISSN = {0001-8708}
}


@book {Hum:1,
    AUTHOR = {Humphreys, James E.},
     TITLE = {Introduction to {L}ie algebras and Representation Theory},
      NOTE = {Second printing, revised},
 PUBLISHER = {Springer-{V}erlag},
   ADDRESS = {New York},
      YEAR = {1978},
    series = {Graduate Texts in Mathematics},
    volume = 9,
      ISBN = {0-387-90053-5}
}


@book{Jac:1,
    author = {Jacobson, Nathan},
     title = {{L}ie Algebras},
 publisher = {Dover},
   address = {New York},
      year = 1979,
 booktitle = {{L}ie Algebras},
      isbn = {0-486-63832-4}
}


@book {Jac:2,
    AUTHOR = {Jacobson, Nathan},
     TITLE = {Basic algebra, {II}},
   EDITION = {Second},
 PUBLISHER = {W. H. Freeman and Company},
   ADDRESS = {New York},
      YEAR = {1989},
     PAGES = {xviii+686},
      ISBN = {0-7167-1933-9}
}


@book{Jan:1,
    AUTHOR = {Jantzen, Jens Carsten},
     TITLE = {Lectures on quantum groups},
 PUBLISHER = {American Mathematical Society},
   ADDRESS = {Providence, RI},
      YEAR = {1996},
     PAGES = {viii+266},
      ISBN = {0-8218-0478-2},
   MRCLASS = {17B37 (16W30)},
  MRNUMBER = {96m:17029},
MRREVIEWER = {Arun Ram},
}


@book {Kas:1,
    AUTHOR = {Kassel, Christian},
     TITLE = {Quantum groups},
 PUBLISHER = {Springer-Verlag},
   ADDRESS = {New York},
      YEAR = {1995},
     PAGES = {xii+531},
      ISBN = {0-387-94370-6},
   MRCLASS = {17B37 (16W30 18D10 20F36 57M25 81R50)},
  MRNUMBER = {96e:17041},
MRREVIEWER = {Yu. N. Bespalov},
}


@book{Lam:1,
    author = {Lam, T.Y.},
     title = {A First Course in Noncommutative Rings},
 publisher = {Springer-{V}erlag},
   address = {New York},
      year = 1991,
    series = {Graduate Texts in Mathematics},
    volume = 131,
 booktitle = {A First Course in Noncommutative Rings},
      isbn = {0-387-97523-3}
}


@book {Lam:2,
    AUTHOR = {Lam, T. Y.},
     TITLE = {Lectures on Modules and Rings},
 PUBLISHER = {Springer-Verlag},
   ADDRESS = {New York},
      YEAR = {1999},
    series = {Graduate Texts in Mathematics},
    volume = 189,
     PAGES = {xxiv+557},
      ISBN = {0-387-98428-3}
}


@article {New:1,
    AUTHOR = {Newman, M. H. A.},
     TITLE = {On Theories with a Combinatorial Definition of
              ``Equivalence.''},
   JOURNAL = {Ann. of Math. (2)},
    VOLUME = {43},
      YEAR = {1942},
     PAGES = {223--243}
}


@article {Ore:1,
    AUTHOR = {Ore, Oystein},
     TITLE = {Theory of Non-commutative Polynomials},
   JOURNAL = {Ann. of Math.},
  FJOURNAL = {Annals of Mathematics},
    VOLUME = {34},
      YEAR = {1933},
     PAGES = {480--508}
}


@book{Reu:1,
    author = {Reutenauer, Christophe},
     title = {Free {L}ie Algebras},
 publisher = {Oxford University Press},
   address = {New York},
      year = 1993,
    series = {London {M}athematical {S}ociety Monographs, New Series},
    volume = 7,
 booktitle = {Free {L}ie Algebras},
      isbn = {0-19-853679-8}
}


@article {Reu:2,
    AUTHOR = {Reutenauer, Christophe},
     TITLE = {Inversion Height in Free Fields},
   JOURNAL = {Selecta Math. (N.S.)},
  FJOURNAL = {Selecta Mathematica. New Series},
    VOLUME = {2},
      YEAR = {1996},
    NUMBER = {1},
     PAGES = {93--109},
      ISSN = {1022-1824}
}


@book {Rot:1,
    AUTHOR = {Rotman, Joseph J.},
     TITLE = {An Introduction to Homological Algebra},
 PUBLISHER = {Academic Press Inc.},
   ADDRESS = {New York},
      YEAR = {1979},
    series = {Pure and Applied Mathematics},
    number = 85,
     PAGES = {xi+376},
      ISBN = {0-12-599250-5}
}      

      
@book {Ser:1,
    AUTHOR = {Serre, Jean-Pierre},
     TITLE = {Complex Semisimple {L}ie Algebras},
      NOTE = {Translated from the French by G. A. Jones,
              Reprint of the 1987 edition},
 PUBLISHER = {Springer-Verlag},
   ADDRESS = {Berlin},
      YEAR = {2001},
     PAGES = {x+74},
      ISBN = {3-540-67827-1}
}


@book {Ste:1,
    AUTHOR = {Stewart, Ian},
     TITLE = {Lie Algebras},
 PUBLISHER = {Springer-Verlag},
   ADDRESS = {Berlin},
      YEAR = {1970},
    series = {Lecture Notes in Mathematics},
    volume = 127,
     PAGES = {iv+97}
}


@book {Swe:1,
    AUTHOR = {Sweedler, Moss E.},
     TITLE = {Hopf Algebras},
 PUBLISHER = {W. A. Benjamin, Inc.},
   address = {New York},
      YEAR = {1969},
    series = {Mathematics Lecture Notes Series},
     PAGES = {vii+336}
}


@incollection {Taf:1,
    AUTHOR = {Taft, Earl J.},
     TITLE = {Quantum Deformation of the Flag Variety},
 BOOKTITLE = {Quantum Groups (Leningrad, 1990)},
     PAGES = {138--141},
 PUBLISHER = {Springer},
   ADDRESS = {Berlin},
      YEAR = {1992},
    series = {Lecture Notes in Mathematics},
    volume = 1510
}


@article {TafTow:1,
    AUTHOR = {Taft, Earl and Towber, Jacob},
     TITLE = {Quantum Deformation of Flag Schemes and {G}rassmann Schemes,
             {I}. {A} $q$-Deformation of the Shape-algebra for ${\rm
              {GL}}(n)$},
   JOURNAL = {J. Algebra},
  FJOURNAL = {Journal of Algebra},
    VOLUME = {142},
      YEAR = {1991},
    NUMBER = {1},
     PAGES = {1--36},
      ISSN = {0021-8693}
}


@book {Wei:1,
    AUTHOR = {Weil, Andr{\'e}},
     TITLE = {Basic Number Theory},
      NOTE = {Reprint of the second (1973) edition},
 PUBLISHER = {Springer-Verlag},
   ADDRESS = {Berlin},
      YEAR = {1995},
    series = {Classics in Mathematics},
    volume = 144,
      ISBN = {3-540-58655-5}
}




