Gradings, graded identities, ⁎-identities and graded ⁎-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 $a_1$, …, $a_n\in A$.

Consider $A$ the subalgebra of $U{T}_3\left(K\right)$ given by:$A=K(e_{1,1}+e_{3,3})\oplus K{e}_{2,2}\oplus K{e}_{2,3}\oplus K{e}_{3,2}\oplus K{e}_{1,3},$ where $e_{i,j}$ denote the matrix units. We investigate the gradings on the algebra $A$, determined by an abelian group, and prove that these gradings are elementary. Furthermore, we compute a basis for the $Z_2$-graded identities of $A$, and also for the $Z_2$-graded identities with graded involution. Moreover, we describe the cocharacters of this algebra.