* -polynomial identities of n ×n matrices: the low degrees.
by
Alain D'Amour
Denison University
Coauthors: Michel Racine, University of Ottawa

Let A be an associative algebra with involution * over a field F . Amitsur defined a * -polynomial to be a polynomial where the variables can also appear with a * ; such polynomials can be evaluated on associative algebras with involution in the obvious way. Unlike ordinary identities, * -identities are not well-understood; for instance, an Amitsur-Levitzki type of result has not yet been established. When A = M_n(F), the n ×n matrices with entries in F, the ideal I(M_n(F), * ) of * - identities coincides with either I(M_n(F), t) for t the transpose involution or I(M_n(F), s) for s the symplectic involution (n even). In both cases, the question of what is the minimal degree of a * -polynomial identity is still open. We investigate the case where n < 5.

Date received: November 16, 1998