arXiv subjects
Publications and source records attributed to Diogo Diniz.
Let $G$ be an arbitrary group and let $\mathbb{F}$ be a finite field. In this paper, we determine bases for the $T_G$-ideals of graded polynomial identities of the algebra $M_2(\mathbb{F})$ for all possible $G$-gradings. The bases obtained consist of finitely many non-trivial graded identities, and are finite whenever $G$ is finite.
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 study the algebra of upper triangular matrices endowed with a group grading and a homogeneous involution over an infinite field. We compute the asymptotic behaviour of its (graded) star-codimension sequence. It turns out that the asymptotic growth of the sequence is independent of the grading and the involution under consideration, depending solely on the size of the matrix algebra. This independence of the group grading also applies to the graded codimension sequence of the associative 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 $3$.
In this paper we classify, up to isomorphism, the superinvolutions on algebras of upper block-triangular matrices over an algebraically closed field of characteristic different from $2$.
In this note we introduce a new technique to answer an issue posed in [7] concerning geometric properties of the set of non-surjective linear operators. We also extend and improve a related result from the same paper.
We investigate algebraic structures within sets of surjective and injective linear operators between sequence spaces, completing results of Aron et al.
We determine the number of isomorphism classes of elementary gradings by a finite group on an algebra of upper block-triangular matrices. As a consequence we prove that, for a finite abelian group $G$, the sequence of the numbers $E(G,m)$ of isomorphism classes of elementary $G$-gradings on the algebra $M_{m}(\mathbb{F})$ of $m\times m$ matrices with entries in a field $\mathbb{F}$ characterizes $G$. A formula for the number of isomorphism classes of gradings by a finite abelian group on an algebra of upper block-triangular matrices over an algebraically closed field, with mild restrictions on its characteristic, is also provided. Finally, if $G$ is a finite abelian group, $\mathbb{F}$ is an algebraically closed field and $N(G,m)$ is the number of isomorphism classes of $G$-gradings on $M_{m}% (\mathbb{F})$ we prove that $N(G,m)\sim\frac{1}{\left\vert G\right\vert !}m^{\left\vert G\right\vert -1}\sim E(G,m)$.
Let $F$ be a field of characteristic zero, $G$ be a group and $R$ be the algebra $M_n(F)$ with a $G$-grading. Bahturin and Drensky proved that if $R$ is an elementary and the neutral component is commutative then the graded identities of $R$ follow from three basic types of identities and monomial identities of length $\geq 2$ bounded by a function $f(n)$ of $n$. In this paper we prove the best upper bound is $f(n)=n$, more generally we prove that all the graded monomial identities of an elementary $G$-grading on $M_n(F)$ follow from those of degree at most $n$. We also study gradings which satisfy no monomial identities but the trivial ones, which we call almost non-degenerate gradings. The description of non-degenerate elementary gradings on matrix algebras is reduced to the description of non-degenerate elementary gradings on matrix algebras that have commutative neutral component. We provide necessary conditions so that the grading on $R$ is almost non-degenerate and we apply the results on monomial identities to describe all almost non-degenerate $\mathbb{Z}$-gradings on $M_n(F)$ for $n\leq 5$.
Let $\mathbb F$ be an algebraically closed field, $G$ be an abelian group, and let $A$ and $B$ be arbitrary finite-dimensional $G$-graded simple algebras over $\mathbb F$. We prove that $A$ and $B$ are isomorphic if, and only if, they satisfy the same graded polynomial identities.
Let $G$ be an abelian group and $\mathbb{K}$ an algebraically closed field of characteristic zero. A. Valenti and M. Zaicev described the $G$-gradings on upper block-triangular matrix algebras provided that $G$ is finite. We prove that their result holds for any abelian group $G$: any grading is isomorphic to the tensor product $A\otimes B$ of an elementary grading $A$ on an upper block-triangular matrix algebra and a division grading $B$ on a matrix algebra. We then consider the question of whether graded identities $A\otimes B$, where $B$ is an algebra with a division grading, determine $A\otimes B$ up to graded isomorphism. In our main result, Theorem 3, we reduce this question to the case of elementary gradings on upper block-triangular matrix algebras which was previously studied by O. M. Di Vincenzo and E. Spinelli.
We describe the graded isomorphisms of rings of endomorphisms of graded flags over graded division algebras. As a consequence describe the isomorphism classes of upper block triangular matrix algebras (over an algebraically closed field of characteristic zero) graded by a finite abelian group.
Let $A$ be a finite dimensional real algebra with a division grading by a finite abelian group $G$. In this paper we provide finite basis for the $T_G$-ideal of graded identities and for the $T_G$-space of graded central polynomials for $A$.
Let $F$ be an infinite field. The primeness property for central polynomials of $M_n(F)$ was proved by A. Regev, i.e., if the product of two polynomials in distinct variables is central then each factor is also central. In this paper we consider the analogous property for $M_n(F)$ and determine, within the elementary gradings with commutative neutral component, the ones that satisfy this property, namely the crossed product gradings. Next we consider $M_n(R)$, where $R$ admits a regular grading, with a grading such that $M_n(F)$ is a homogeneous subalgebra and provide sufficient conditions - satisfied by $M_n(E)$ with the trivial grading - to prove that $M_n(R)$ has the primeness property if $M_n(F)$ does. We also prove that the algebras $M_{a,b}(E)$ satisfy this property for ordinary central polynomials. Hence over a field of characteristic zero every verbally prime algebra as the primeness property.
Let $K$ be a field of characteristic zero and $V$ a vector space of dimension $m>1$ with a nondegenerate symmetric bilinear form $f:V\times V \rightarrow K$. The Jordan algebra $B_m=K\oplus V$ of the form $f$ is a superalgebra with this decomposition. We prove that the ideal of all the $2$-graded identities of $B_m$ satisfies the Specht property and we compute the $2$-graded cocharacter sequence of $B_m$.
The Bohnenblust-Hille inequality was obtained in 1931 and (in the case of real scalars) asserts that for every positive integer $N$ and every $m$-linear mapping $T:\ell_{\infty}^{N}\times...\times\ell_{\infty}^{N}\rightarrow \mathbb{R}$ one has (\sum\limits_{i_{1},...,i_{m}=1}^{N}|T(e_{i_{^{1}}},...,e_{i_{m}})|^{\frac{2m}{m+1}})^{\frac{m+1}{2m}}\leq C_{m}\VertT\Vert, for some positive constant $C_{m}$. Since then, several authors obtained upper estimates for the values of $C_{m}$. However, the novelty presented in this short note is that we provide lower (and non-trivial) bounds for $C_{m}$.
We provide (for both the real and complex settings) a family of constants, $% (C_{m})_{m\in \mathbb{N}}$, enjoying the Bohnenblust--Hille inequality and such that $\displaystyle\lim_{m\rightarrow \infty}\frac{C_{m}}{C_{m-1}}=1$, i.e., their asymptotic growth is the best possible. As a consequence, we also show that the optimal constants, $(K_{m})_{m\in \mathbb{N}}$, in the Bohnenblust--Hille inequality have the best possible asymptotic behavior. Besides its intrinsic mathematical interest and potential applications to different areas, the importance of this result also lies in the fact that all previous estimates and related results for the last 80 years (such as, for instance, the multilinear version of the famous Grothendieck Theorem for absolutely summing operators) always present constants $C_{m}$'s growing at an exponential rate of certain power of $m$.