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Lucio Centrone

Publications and source records attributed to Lucio Centrone.

At least 19 recordsLinked to original sources
arXiv2026

Quantum regularity of finite dimensional semisimple algebras

In this paper, we establish a necessary and sufficient criterion for a finite-dimensional semisimple algebra over an algebraically closed field of characteristic $0$ to admit a regular quantum commutative decomposition. We apply this characterization to group algebras and show that a finite group algebra admits such a decomposition if and only if the underlying group is a peak group, that is, a finite group whose irreducible character degrees have a greatest element with respect to the divisibility ordering. We investigate the structure of peak groups and establish several criteria for their solvability in terms of the prime divisors of their largest irreducible character degree. In particular, we show that several important classes of groups are peak groups, while supersolvability alone does not imply the peak property. Finally, we construct a non-solvable peak group within the class of Frobenius groups.

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arXiv2026

Images of polynomials with involution on $2\times 2$ matrices

Let $\mathbb{F}$ be a field and let $M_2(\mathbb{F})$ be the algebra of $2\times 2$ matrices endowed with an involution of the first kind. We study the image of multilinear $*$-polynomials evaluated on $M_2(\mathbb{F})$. For the transpose involution over $\mathbb{R}$, we show that the image is either a proper vector subspace or contains a basis of $M_2(\mathbb{R})$. For the symplectic involution over quadratically closed fields or over $\mathbb{R}$, we prove that the image is always a vector space, namely one of $\{0\}$, $\mathbb{F}$, $sl_2(\mathbb{F})$ or $M_2(\mathbb{F})$. As a byproduct, we complete a theorem of Bre\v{s}ar and Klep describing the linear span of the image of a $*$-polynomial on finite dimensional central simple algebras with involution of the first kind. Their result excluded algebras of dimensions 4 and 16; we settle both cases, extending the description to all dimensions greater than 1 (over $\mathbb{R}$ for the transpose involution, and over quadratically closed fields or $\mathbb{R}$ for the symplectic involution). We also classify all Lie skew-ideals of $M_4(\mathbb{F})$ over fields of characteristic zero.

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arXiv2026

On Regular Quantum Commutative Algebras

Let $K$ be an algebraically closed field of characteristic different from $2$. We provide a positive solution to the Bahturin--Regev conjecture in the general finite-dimensional (non-graded) setting, assuming that $\operatorname{char}(K)$ does not divide the quantum length of a minimal regular quantum commutative decomposition. Furthermore, we obtain a criterion, formulated in terms of regular quantum commutative decompositions, under which a set-grading on a semisimple associative algebra is realized as a group grading.

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arXiv2026

Primeness property for regular gradings

Let $K$ be an algebraically closed field of characteristic $0$ and $G$ a finite abelian group. For a $G$-graded $K$-algebra $A$, we define the primeness property for graded central polynomials: for any graded polynomials $f$ and $g$ in disjoint sets of variables, if $fg$ is graded central, then both $f$ and $g$ are graded central. Let $A=\bigoplus_{g\in G} A_g$ be its decomposition into homogeneous components. Assume that for every $n$-tuple $(g_1,\dots,g_n)$ in $G$, there exist $a_{i}\in A_{g_{i}}$ with $a_1\cdots a_n\neq 0$, and that for each $g$,$h\in G$ there exists a scalar $\beta(g,h)\in K^{\ast}$ such that $a_ga_h=\beta(g,h)a_ha_g$. Then the grading is regular, and minimal if no distinct $g$, $h\in G$ satisfy $\beta(g,x)=\beta(h,x)$ for all $x\in G$. We prove that $G$-graded regular algebras, including $M_n(K)$ with the Pauli grading, fail the primeness property. For matrices of orders $2$ and $3$, no nontrivial gradings satisfy primeness. Finally, for $\mathbb{Z}_2$-graded regular algebras, we use the known fact that minimal regular gradings satisfy the graded identities of the infinite-dimensional Grassmann algebra $E$ and contain a copy of $E$ to show that such algebras satisfy the primeness property in the ordinary sense. As a consequence, we show that minimality is not required for the regularity of the grading.

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arXiv2025

On infinite dimensional algebras with regular gradings

Let $G$ be a finite abelian group and let $K$ be an algebraically closed field of characteristic 0. We consider associative unital algebras $A$ over $K$ graded by $G$, that is $A=\oplus_{g\in G} A_g$, where the vector subspaces $A_g$ satisfy $A_gA_h\subseteq A_{g+h}$ for every $g$, $h\in G$. Such a $G$-grading is called regular whenever for every $n$-tuple $(g_1,\ldots,g_n)\in G^n$ there exist homogeneous elements $a_i\in A_{g_i}$ such that $a_1\cdots a_n\ne 0$ in $A$; furthermore, for every $g$, $h\in G$ and every $a_g\in A_g$, $a_h\in A_h$ one has $a_ga_h=\beta(g,h)a_ha_g$ for some $\beta(g,h)\in K^*$. Here $\beta(g,h)$ depends only on the choice of $g$ and $h$ but not on the elements $a_g$ and $a_h$. It is immediate that $\beta$ is a bicharacter on $G$. The regular decomposition above is minimal if for every $g\in G$ with $\beta(g,h)=\beta(g,k)$ one has $h=k$. In this paper we prove that if $G=\mathbb{Z}_2$ then every $G$-graded regular algebra whose regular decomposition is minimal, contains a copy of the infinite dimensional Grassmann algebra. By applying this result we are able to describe the generating algebras of the variety of $\mathbb{Z}_2$-graded algebras defined by the Grassmann algebra. Furthermore we describe the finitely generated graded subalgebras of a $\mathbb{Z}_2$-graded regular algebra having a minimal regular decomposition.

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arXiv2025

Embedding theorems as a bridge between supertraces and supergeometry

Any algebra herein is intended over a field of characteristic 0. Let $E$ denote the infinite dimensional Grassman algebra. Given a power associative finite dimensional {$\mathbb{Z}_2$-graded-central-simple} $A$ and a supertrace algebra $B$, so that $B$ belongs to the same variety of $A\otimes E$, we study conditions on $B$ so that it can be embedded into $A\otimes\Xi$, where $\Xi$ is a supercommutative algebra, called $A$-universal supermap of $B$, provided $B$ satisfies all the supertrace identities of $A\otimes E$. We use this result in order to relate the formal smoothness of $B$ with that of its $A$-universal supermap.

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arXiv2025

On finite dimensional regular gradings

Let $A$ be an associative algebra over an algebraically closed field $K$ of characteristic 0. A decomposition $A=A_1\oplus\cdots \oplus A_r$ of $A$ into a direct sum of $r$ vector subspaces is called a \textsl{regular decomposition} if, for every $n$ and every $1\le i_j\le r$, there exist $a_{i_j}\in A_{i_j}$ such that $a_{i_1}\cdots a_{i_n}\ne 0$, and moreover, for every $1\le i,j\le r$ there exists a constant $\beta(i,j)\in K^*$ such that $a_ia_j=\beta(i,j)a_ja_i$ for every $a_i\in A_i$, $a_j\in A_j$. We work with decompositions determined by gradings on $A$ by a finite abelian group $G$. In this case, the function $\beta\colon G\times G\to K^*$ ought to be a bicharacter. A regular decomposition is {minimal} whenever for every $g$, $h\in G$, the equalities $\beta(x,g)=\beta(x,h)$ for every $x\in G$ imply $g=h$. In this paper we describe completely the structure of the finite dimensional algebras $A$ (with unit) admitting a $G$-regular grading. Moreover, we compute the graded codimension sequence for a class of such algebras assuming complete support and minimal regular decomposition. It turns out that, for these algebras, the graded PI-exponent coincides with the ordinary (ungraded) PI-exponent. Finally, we show that the regular decomposition of a finite-dimensional algebra $A$ with a regular $G$-grading is minimal if and only if $\exp(A)=|G|$.

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arXiv2025

On the Nowicki Conjecture for the free Lie algebra of rank 2

Let K[X_n]=K[x_1,\ldots,x_n] be the polynomial algebra in n variables over a field K of characteristic zero. A locally nilpotent linear derivation \delta of K[X_n] is called Weitzenb\"ock due to his well known result from 1932 stating that the algebra \text{\rm ker}(\delta)=K[X_n]^{\delta} of constants of $\delta$ is finitely generated. The explicit form of a generating set of $K[X_n,Y_n]^{\delta}$ was conjectured by Nowicki in 1994 in the case \delta was such that \delta(y_{i})=x_{i}$, $\delta(x_i)=0, i=1,\ldots,n. Nowicki's conjecture turned out to be true and, recently, has been applied to several relatively free associative algebras. In this paper, we consider the free Lie algebra \mathcal{L}(x,y) of rank 2 generated by x and y over K and we assume the Weitzenb\"ock derivation \delta sending y to x, and x to zero. We introduce the idea of pseudodeterminants and we present a characterization of Hall monomials that are constants showing they are not so far from being pseudodeterminants. We also give a complete list of generators of the constants of degree less than 7 which are, of course, pseudodeterminants.

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arXiv2024

A negative answer to a Bahturin-Regev conjecture about regular algebras in positive characteristic

Let $A=A_1\oplus\cdots\oplus A_r$ be a decomposition of the algebra $A$ as a direct sum of vector subspaces. If for every choice of the indices $1\le i_j\le r$ there exist $a_{i_j}\in A_{i_j}$ such that the product $a_{i_1}\cdots a_{i_n}\ne 0$, and for every $1\le i,j\le r$ there is a constant $\beta(i,j)\ne 0$ with $a_ia_j=\beta(i,j) a_ja_i$ for $a_i\in A_i$, $a_j\in A_j$, the above decomposition is regular. Bahturin and Regev raised the following conjecture: suppose the regular decomposition comes from a group grading on $A$, and form the $r\times r$ matrix whose $(i,j)$th entry equals $\beta(i,j)$. Then this matrix is invertible if and only if the decomposition is minimal (that is one cannot get a regular decomposition of $A$ by coarsening the decomposition). Aljadeff and David proved that the conjecture is true in the case the base field is of characteristic 0. We prove that the conjecture does not hold for algebras over fields of positive characteristic, by constructing algebras with minimal regular decompositions such that the associated matrix is singular.

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arXiv2024

Moduli, Deformations and Algebraic K-Theory of Graded PI-Algebras

Over a field of characteristic zero, for a finite grading group and a finitely generated graded $T$-ideal, we construct relatively free sheaves and finite-rank quotient stacks of algebra laws with prescribed identities. Their tangent complexes are PI-restricted Hochschild complexes, and the second variations of the identities give explicit lifting obstructions. The first-order deformation groupoid carries a natural relative algebraic K-theory functor; for its square-zero extensions, the relative Chern character identifies rational relative K-theory with negative cyclic homology. Additional identities determine closed PI-strata and localisation fibre sequences in K-theory with Azumaya coefficients. On the Azumaya substack, the degree annihilates the Brauer class, and Morita transport yields a base-change-compatible K-theory of PI-Azumaya families together with a K-theoretic monodromy equivalence.

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arXiv2023

Cocharacters of $UT_n(E)$

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 $UT_n(E)$.

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arXiv2022

Images of graded polynomials on matrix algebras

The aim of this paper is to start the study of images of graded polynomials on full matrix algebras. We work with the matrix algebra $M_n(K)$ over a field $K$ endowed with its canonical $\mathbb{Z}_n$-grading (Vasilovsky's grading). We explicitly determine the possibilities for the linear span of the image of a multilinear graded polynomial over the field $\mathbb Q$ of rational numbers and state an analogue of the L'vov-Kaplansky conjecture about images of multilinear graded polynomials on $n\times n$ matrices, where $n$ is a prime number. We confirm such conjecture for polynomials of degree 2 over $M_n(K)$ when $K$ is a quadratically closed field of characteristic zero or greater than $n$ and for polynomials of arbitrary degree over matrices of order 2. We also determine all the possible images of semi-homogeneous graded polynomials evaluated on $M_2(K)$.

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arXiv2022

On Nowicki's conjecture: a survey and a new result

The goal of the paper is twofold: it aims to give an extensive set of tools and bibliography towards Nowicki's conjecture both in an associative setting; it establishes a new result about Nowicki's conjecture for the free metabelian Poisson algebra.

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arXiv2020

Graded monomial identities and almost non-degenerate gradings on matrices

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 $n\leq 5$.

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arXiv2018

The Nowicki Conjecture for relatively free algebras

A linear locally nilpotent derivation of the polynomial algebra $K[X_m]$ in $m$ variables over a field $K$ of characteristic 0 is called a Weitzenb\"ock derivation. It is well known from the classical theorem of Weitzenb\"ock that the algebra of constants $K[X_{m}]^{\delta}$ of a Weitzenb\"ock derivation $\delta$ is finitely generated. Assume that $\delta$ acts on the polynomial algebra $K[X_{2d}]$ in $2d$ variables as follows: $\delta(x_{2i})=x_{2i-1}$, $\delta(x_{2i-1})=0$, $i=1,\ldots,d$. The Nowicki conjecture states that the algebra $K[X_{2d}]^{\delta}$ is generated by $x_1,x_3.\ldots,x_{2d-1}$, and $x_{2i-1}x_{2j}-x_{2i}x_{2j-1}$, $1\leq i<j\leq d$. The conjecture was proved by several authors based on different techniques. We apply the same idea to two relatively free algebras of rank $2d$. We give the infinite set of generators of the algebra of constants in the the free metabelian associative algebras $F_{2d}(\mathfrak A)$, and finite set of generators in the free algebra $F_{2d}(\mathcal G)$ in the variety determined by the identities of the infinite dimensional Grassmann algebra.

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arXiv2016

On the Z_2-graded codimensions of the Grassmann algebra over a finite field

Let E be the infinite dimensional Grassmann algebra over a finite field F of characteristic not 2. In this paper we deal with the homogeneous Z_2-gradings of E. In particular, we compute an exact value for the Z_2-graded homogeneous codimensions of E, and a lower and an upper bound for the Z_2-graded (non-homogeneous) codimensions of E for each of its Z_2-homogeneous grading.

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arXiv2015

On the growth of graded polynomial identities of sl_n

Let K be a field of characteristic 0 and L be a G-graded Lie PI-algebra, where G is a finite group. We define the graded Gelfand-Kirillov dimension of L. Then we measure the growth of the Z_n-graded polynomial identities of the Lie algebra of n x n traceless matrices sl_n(K) giving an exact value of its Z_n-graded Gelfand-Kirillov dimension.

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