Identities with involutions on incidence algebras

Abstract

In this paper, we study polynomial identities with involution of an incidence algebra I(P, F) where P is a connected finite poset with an involution λ and F is a field of characteristic zero. At first, we also consider P of length at most 2 and then of length at most 3. Let and denote, respectively, the λ-orthogonal and the λ-symplectic involutions of I(P, F). For the case that P has length at most 2 and , we show that the -identities and the -identities of I(P, F) follow from the ordinary identity . In that context, passing to the particular case , where is a poset called crown with 2n elements, and using the classification of the involutions on , we show that, for all involutions ρ on , every ρ-identity also follows from the ordinary identity . For the case that P has length at most 3 and , we determine the generators of the -ideal when every element of P that is neither minimal nor maximal is fixed by λ and, for such an element, there exists a unique minimal element of P that is comparable with it. Communicated by Igor Klep