Gradings, graded identities, $\ast$-identities and graded $\ast$-identities of an algebra of upper triangular matrices

Abstract

Let $K⟨X⟩$ be the free associative algebra freely generated over the field $K$ by the countable set $X=\{x_1,x_2,\dots \}$. If $A$ is an associative $K$-algebra, we say that a polynomial $f(x_1,\dots ,x_n)\in K⟨X⟩$ is a polynomial identity, or simply an identity in $A$ if $f(a_1,\dots ,a_n)=0$ for every $a_1,\dots ,a_n\in A$. Consider $\mathcal{A}$ the subalgebra of $U{T}_3(K)$ given by: \[ \mathcal{A} = K(e_{1,1} + e_{3,3}) \oplus Ke_{2,2} \oplus Ke_{2,3} \oplus Ke_{3,2} \oplus Ke_{1,3} , \] where $e_{i,j}$ denote the matrix units. We investigate the gradings on the algebra $\mathcal{A}$, determined by an abelian group, and prove that these gradings are elementary. Furthermore, we compute a basis for the ${\mathbb{Z}}_2$-graded identities of $\mathcal{A}$, and also for the ${\mathbb{Z}}_2$-graded identities with graded involution. Moreover, we describe the cocharacters of this algebra.