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 $\widehat{\lambda }$ and ${\sigma }_{\lambda }$ 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 $\left|P\right|\ge 4$, we show that the $\widehat{\lambda }$-identities and the ${\sigma }_{\lambda }$-identities of I(P, F) follow from the ordinary identity $[x_1,x_2][x_3,x_4]$. In that context, passing to the particular case $I(C_{2n},F)$, where $C_{2n}$ is a poset called crown with 2n elements, and using the classification of the involutions on $I(C_{2n},F)$, we show that, for all involutions ρ on $I(C_{2n},F)$, every ρ-identity also follows from the ordinary identity $[x_1,x_2][x_3,x_4]$. For the case that P has length at most 3 and $\left|P\right|\ge 4$, we determine the generators of the $T(\widehat{\lambda })$-ideal $I{d}^{\widehat{\lambda }}(I(P,F))$ 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

Keywords:

• Incidence algebras
• Polynomial identities
• Involution
• Partially ordered set

2020 Mathematics Subject Classification:

• 16R10
• 16R50
• 16W10

Acknowledgments

We thank the referee for the valuable suggestions and comments which improved this paper.

Disclosure statement

The authors report there are no competing interests to declare.

Additional information

Funding

The authors acknowledge with gratitude the support from the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) and Fundação Araucária. The first author was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001. The second author was partially supported by Fundação Araucária, PPP 13/2009, Convênio 232/2010 - 17360.