Involutions of semisimple group algebras

Abstract EN

Our aim in this paper is to determine necessary and sufficient conditions for which: (1) any involution on a finite-dimensional semisimple algebra A leaves invariant each simple component of A.

(2) the canonical involution of K[G] induces an involution of the first kind on each simple component of K[G].

These conditions are obtained in terms of: (i) conjugacy relation in G (Theorem 2).

(k) characters of G and hermitian forms (Theorem 3).

(3) the canonical involution of K[G] induces an involution of the second kind on any simple component of K{G] (Theorem 1).

As an application of these results, we give an improved version of theorem 13.3 of [[4], p.

323] (Theorem 5).

The final section is devoted to investigating a special class of groups used in this work and called (c)-groups.