148 research outputs found

### On the Graded Identities for Elementary Gradings in Matrix Algebras over Infinite Fields

We find a basis for the $G$-graded identities of the $n\times n$ matrix
algebra $M_n(K)$ over an infinite field $K$ of characteristic $p>0$ with an
elementary grading such that the neutral component corresponds to the diagonal
of $M_n(K)$

### Graded identities with involution for the algebra of upper triangular matrices

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 $3$

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