arXiv subjects
Publications and source records attributed to Daniela Martinez Correa.
Let $G$ be a finite group and let $\F$ be a finite field. We prove that any finite-dimensional $G$-graded associative algebra $A$ over $\F$ has a finite basis for its $G$-graded polynomial identities.
We prove that two finite prime $Ω$-algebras defined over the same unital commutative ring and satisfying the same set of polynomial identities are isomorphic.
We compute the graded polynomial identities for the variety of graded algebras generated by the Lie algebra of upper triangular matrices of order 3 over an arbitrary field and endowed with an elementary grading. We investigate the Specht property for the same family of varieties.
Let $F$ be a field of characteristic $0$ and let $E$ be the infinite dimensional Grassmann algebra over $F$. In the first part of this paper we give an algorithm calculating the generating function of the cocharacter sequence of the $n\times n$ upper triangular matrix algebra $UT_n(E)$ with entries in $E$, lying in a strip of a fixed size. In the second part we compute the double Hilbert series $H(E;\mathrm{T}_k,\mathrm{Y}_l)$ of $E$, then we define the $(k,l)$-multiplicity series of any PI-algebra. As an application, we derive from $H(E;\mathrm{T}_k,\mathrm{Y}_l)$ an easy algorithm determining the $(k,l)$-multiplicity series of $UT_n(E)$.
Let $UT_n(F)$ be the algebra of the $n\times n$ upper triangular matrices and denote $UT_n(F)^{(-)}$ the Lie algebra on the vector space of $UT_n(F)$ with respect to the usual bracket (commutator), over an infinite field $F$. In this paper, we give a positive answer to the Specht property for the ideal of the $\mathbb{Z}_n$-graded identities of $UT_n(F)^{(-)}$ with the canonical grading when the characteristic $p$ of $F$ is 0 or is larger than $n-1$. Namely we prove that every ideal of graded identities in the free graded Lie algebra that contains the graded identities of $UT_n(F)^{(-)}$, is finitely based. Moreover we show that if $F$ is an infinite field of characteristic $p=2$ then the $\mathbb{Z}_3$-graded identities of $UT_3^{(-)}(F)$ do not satisfy the Specht property. More precisely, we construct explicitly an ideal of graded identities containing that of $UT_3^{(-)}(F)$, and which is not finitely generated as an ideal of graded identities.