Symmetric and skew-symmetric polynomial identities with involution for the upper triangular matrix algebras of even order

Abstract

Let UT2n be the algebra of all 2n×2n upper triangular matrices over a field F whose characteristic is different from 2. In the present work, we will consider UT2n equipped with an involution of the first kind. In this setting, for certain subalgebras of UT2n (including UT2n itself), we will describe all the ⁎-polynomial identities which are of the form f±f⁎ where f is a product of commutators. In addition, we will exhibit sets of linear generators for the relatively free algebras induced from these types of identities.