Narrow Results

We consider the algebra U_n(K) of n×n upper triangular matrices over an infinite field K equipped with its usual ℤ_n-grading. We describe a basis of the ideal of the graded polynomial identities for this algebra.

The Lie algebra sl₂(K) over a field K has a natural grading by ℤ₂, the cyclic group of order 2. We describe the graded polynomial identities for this grading when the base field is infinite and of characteristic different from 2. We exhibit a basis of these identities that consists of one polynomial. In order to obtain this basis we employ methods and results from Invariant theory. As a by-product we obtain finite bases of the graded identities for sl₂(K) graded by other groups such as ℤ₂ × ℤ₂, and by the integers ℤ.

We consider the infinite dimensional Grassmann algebra E over a field F of characteristic 0 or p, where p > 2, and we compute its ℤ₂-graded Gelfand–Kirillov (GK) dimension as a ℤ₂-graded PI-algebra.

Let $G$ be a finite abelian group. As a consequence of the results of Di Vincenzo and Nardozza, we have that the generators of the $T_G$-ideal of $G$-graded identities of a $G$-graded algebra in characteristic 0 and the generators of the $T_{G\times {\mathbb{Z}}_2}$-ideal of $G\times {\mathbb{Z}}_2$-graded identities of its tensor product by the infinite-dimensional Grassmann algebra $E$ endowed with the canonical grading have pairly the same degree. In this paper, we deal with ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded identities of $E_{k^{\ast }}\otimes E$ over an infinite field of characteristic $p>2$, where $E_{k^{\ast }}$ is $E$ endowed with a specific ${\mathbb{Z}}_2$-grading. We find identities of degree $p+1$ and $p+2$ while the maximal degree of a generator of the ${\mathbb{Z}}_2$-graded identities of $E_{k^{\ast }}$ is $p$ if $p>k$. Moreover, we find a basis of the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded identities of $E_{k^{\ast }}\otimes E$ and also a basis of multihomogeneous polynomials for the relatively free algebra. Finally, we compute the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Gelfand–Kirillov (GK) dimension of $E_{k^{\ast }}\otimes E$.

Let K be a field of characteristic zero and let ${𝔰𝔩}_2(K)$ be the 3-dimensional simple Lie algebra over K. In this paper, we describe a finite basis for the ${\mathbb{Z}}_2$-graded identities of the adjoint representation of ${𝔰𝔩}_2(K)$, or equivalently, the ${\mathbb{Z}}_2$-graded identities for the pair $(M_3(K),{𝔰𝔩}_2(K))$. We work with the canonical grading on ${𝔰𝔩}_2(K)$ and the only nontrivial ${\mathbb{Z}}_2$-grading of the associative algebra $M_3(K)$ induced by that on ${𝔰𝔩}_2(K)$.