Graded identities with involution for the algebra of upper triangular matrices

TL;DR

This paper classifies graded involutions on upper triangular matrix algebras over characteristic zero fields, providing bases for identities, growth rates, and exponents, revealing the structure and complexity of these algebraic objects.

Contribution

It characterizes gradings with involution on upper triangular matrices, establishes bases for identities, and determines asymptotic growth and exponents, advancing understanding of graded algebra structures.

Findings

Graded involutions are coarsenings of specific $bZ^{loor{rac{m}{2}}}$-gradings.

Explicit bases for identities under reflection and symplectic involutions are provided.

The $(G,\ast)$-exponent is determined to be $m$ or $m+1$ depending on parity.

Abstract

Let $F$ be a field of characteristic zero. We prove that if a group grading on $UT_m(F)$ admits a graded involution then this grading is a coarsening of a $\mathbb{Z}^{\lfloor\frac{m}{2}\rfloor}$-grading on $UT_m(F)$ and the graded involution is equivalent to the reflection or symplectic involution on $UT_m(F)$. A finite basis for the $(\mathbb{Z}^{\lfloor\frac{m}{2}\rfloor},\ast)$-identities is exhibited for the reflection and symplectic involutions and the asymptotic growth of the $(\mathbb{Z}^{\lfloor\frac{m}{2}\rfloor},\ast)$-codimensions is determined. As a consequence we prove that for any $G$-grading on $UT_m(F)$ and any graded involution the $(G,\ast)$-exponent is $m$ if $m$ is even and either $m$ or $m+1$ if $m$ is odd. For the algebra $UT_3(F)$ there are, up to equivalence, two non-trivial gradings that admit a graded involution: the canonical $\mathbb{Z}$-grading and the $\mathbb{Z}_2$-grading induced by $(0,1,0)$. We determine a basis for the $(\mathbb{Z}_2,\ast)$-identities and prove that the exponent is $3$. Hence we conclude that the ordinary $\ast$-exponent for $UT_3(F)$ is $3$.