Involutions and symmetric elements in group algebras
Speaker: Dr. Zsolt Balogh (Institute of Mathematics and Informatics, Hungary)
Date: Wed 27th January 2010
Location: Mathematical Sciences Seminar Room
Let A be an algebra with an involution *. An element x in A is called symmetric (skew symmetric) with respect to *, if x^*=x (x^*=-x). Denote A^+ and A^- the set of symmetric and skew symmetric elements of A, respectively. Amitsur proved that an algebra satisfies a polynomial identity if and only if the set of symmetric elements of the algebra satisfies a polynomial identity. We remark that the polynomial identity which is satisfied by the algebra is not necessarily the same as the one which is satisfied by the symmetric elements.
Let FG be a group algebra of a group G over a field F of characteristic p, and let * be an involution of FG. We would like to give a survey concerning the theory of polynomial identities and group identities in the set of symmetric (skew symmetric) elements of FG. Under the canonical involution of FG we give some identities in FG^+ (FG^-), which can be transferred to the whole algebra. Furthermore, for some Lie-identities we consider not only the canonical involutions.
(This talk is part of the K-Theory, Quadratic Forms and Number Theory series.)