Central Polynomials with Involution of $M_{1,1}(E)$

TL;DR

This paper investigates the structure of central polynomials with involution in the algebra of 2x2 matrices over an infinite field, focusing on those with involutions derived from superinvolutions on matrix algebras.

Contribution

It characterizes the $*$-space of central polynomials with involution in $M_{1,1}(E)$, extending understanding of polynomial identities with involution in superalgebra contexts.

Findings

Description of the $*$-space $C(R,*)$ for $R= M_{1,1}(E)$

Connection between involutions and superinvolutions on matrix algebras

New insights into central polynomials in superinvolution settings

Abstract

Let $K$ be an infinite field of characteristic $\neq 2$. In this article we study the $*$-space $C(R,*)$ of central polynomials with involution of the $K$-algebra $R= M_{1,1}(E)$, with an involution ($*$) obtanied from a superinvolution on $M_{1,1}(F)$ (i.e. $M_2(F)$ with its canonical $\mathbb{Z}_2$-grading).