OHIO UNIVERSITY CENTER FOR
RING THEORY AND ITS APPLICATIONS
LECTURE SERIES
Idempotents, Unitary Units, and Skew Elements in Group Algebras
and Some Applications
Monday,
Nov 14th, 2005
Time:
6:00 PM
Morton
Hall, Room 326
Abstract:
We shall start in a very elementary way, recalling some of the history of
group algebras and early results on
central idempotents. Then, we shall sketch a recent proof of the Theorem of
Berman-Witt based on the structure of the group algebra rather
than character
theory. Finally, we shall survey some recent constructions of
central
idempotents of rational and finite group algebras from the lattice
of subgroups
of the given group.
Tuesday,
Nov 15th, 2005
Time:
7:00 PM
Morton
Hall, Room 122
Abstract: First, we shall show
that if A is an abelian group and F a field such
that FG is
semisimple, then the number of simple components of FA is minimal when
F = Q, the field of
rational numbers. We shall give conditions on a finite field F for
the number
of simple components of FA to be equal to that of QA. Under these
conditions,
we can use a construction of the set of primitive idempotents to
describe all
minimal cyclic and abelian codes, extending some classical results
of S.D.
Berman.
(Under the
auspices of OU-OSU Ring Theory Seminar)
Friday,
Nov 18th, 2005
Time:
4:45 PM
OSU, Math Tower, Room MW154
Abstract:
A group algebra can be viewed, in a natural way, as a ring with
involution. We shall
consider first
the group of unitary units of this ring and describe when if
contains a free group of rank 2 or, otherwise, when it satisfies a group identity.
This last
problem is related to the Lie nilpotence and the commutativity of
the Lie algebra of skew elements, which will also be discussed.