arXiv subjects

Carla Rizzo

Publications and source records attributed to Carla Rizzo.

9 recordsLinked to original sources

arXiv2026

Differential varieties of upper triangular matrices

Let $L$ be a Lie algebra acting by derivations on an associative algebra $A$ over a field $F$ of characteristic zero. The polynomial identities satisfied by $A$ with respect to this action are called differential identities, or $L$-identities. In this paper, we study the differential identities of the algebra $UT_k$ of $k\times k$ upper triangular matrices and take a first step toward the classification of minimal $L$-varieties of differential exponent $3$. We first prove that, whenever $UT_k$ generates a minimal variety of algebras with derivations, the $L$-action can be replaced by its semisimple part. More precisely, it is enough to consider inner derivations induced by diagonal elements. We then apply this reduction to $UT_3$ and explicitly determine the $T_L$-ideal of differential identities and the corresponding differential codimension sequence for every such action on $UT_3$. Finally, we show that every $L$-variety generated by $UT_k$, with $k\geq 3$, contains $UT_3$ endowed with one of these $L$-actions.

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arXiv2026

On anti-left-invariant and left-invariant pseudo-Euclidean nearly associative algebras

We introduce the notions of anti-left-invariant and left-invariant pseudo-Euclidean nearly associative algebras, which arise as nearly associative Levi-Civita products associated with pseudo-Euclidean Lie and Jordan algebras, respectively. We establish a correspondence between these classes of algebras and the Levi-Civita products of their associated Lie and Jordan structures. For anti-left-invariant pseudo-Euclidean nearly associative algebras, we prove that they are nilpotent of index at most five and characterize them as Jacobi--Jordan-admissible nearly associative algebras. We further show that the associated pseudo-Euclidean Jacobi--Jordan algebras are cyclic. Motivated by the classical double extension of Medina and Revoy, we introduce a double extension procedure for this class of algebras and prove that every anti-left-invariant pseudo-Euclidean nearly associative algebra can be obtained from a trivial pseudo-Euclidean algebra by a finite sequence of such extensions. For left-invariant pseudo-Euclidean nearly associative algebras, we prove that the associated pseudo-Euclidean Lie algebras are two-step solvable and cyclic. We then develop block, planar, and linear double extensions and show that every left-invariant pseudo-Euclidean nearly associative algebra can be recursively constructed from a quadratic commutative associative algebra by means of block double extensions. Moreover, we prove that over the field of real numbers, every such algebra can be recursively constructed from a quadratic commutative associative algebra using planar double extensions. These recursive constructions provide a unified framework for describing and classifying pseudo-Euclidean nearly associative algebras in both the anti-left-invariant and left-invariant settings.

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arXiv2025

Varieties of graded $W$-algebras and asymptotic behavior of codimension growth

Let $W$ be a $G$-graded algebra over a field of characteristic zero, where $G$ is a finite group. We develope a theory of generalized $G$-graded polynomial identities satisfied by any finite-dimensional $W$-algebra $A$, by mean of the graded multiplier algebra of $A.$ In particular, we first prove that the graded generalized exponent exists and equals the ordinary one. Then, we explicitly compute the $G$-graded generalized identities of $UT_2,$ the $2 \times 2$ upper triangular matrix algebra equipped with its canonical $\mathbb{Z}_2$-grading, under all the possible graded $W$-actions. Finally, we exhibit examples of varieties of graded $W$-algebras with almost polynomial growth of the codimensions.

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arXiv2025

Multipliers, $W$-algebras and the growth of generalized polynomial identities

Let $A$ be a $W$-algebra over a field $F$ of characteristic zero, where $W$ is any $F$-algebra. We first develop a comprehensive theory of generalized identities independent of the algebraic structure of $W$, using the multiplier algebra of $A.$ Then, we investigate the generalized variety generated by the $k\times k$ matrix algebra with a suitable action, proving that it exhibits almost polynomial growth of the generalized codimensions. Furthermore, we characterize the generalized varieties of almost polynomial growth generated by finite dimensional $W$-algebras. Finally, we provide a counterexample to the Specht property of generalized $T_W$-ideals in characteristic zero.

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arXiv2024

Bialgebra theory for nearly associative algebras and $LR$-algebras: equivalence, characterization, and $LR$-Yang-Baxter Equation

We develop the bialgebra theory for two classes of non-associative algebras: nearly associative algebras and $LR$-algebras. In particular, building on recent studies that reveal connections between these algebraic structures, we establish that nearly associative bialgebras and $LR$-bialgebras are, in fact, equivalent concepts. We also provide a characterization of these bialgebra classes based on the coproduct. Moreover, since the development of nearly associative bialgebras - and by extension, $LR$-bialgebras - requires the framework of nearly associative $L$-algebras, we introduce this class of non-associative algebras and explore their fundamental properties. Furthermore, we identify and characterize a special class of nearly associative bialgebras, the coboundary nearly associative bialgebras, which provides a natural framework for studying the Yang-Baxter equation (YBE) within this context.

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arXiv2024

Differential identities of matrix algebras

We study the differential identities of the algebra $M_k(F)$ of $k\times k$ matrices over a field $F$ of characteristic zero when its full Lie algebra of derivations, $L=\mbox{Der}(M_k(F))$, acts on it. We determine a set of 2 generators of the ideal of differential identities of $M_k(F)$ for $k\geq 2$. Moreover, we obtain the exact values of the corresponding differential codimensions and differential cocharacters. Finally we prove that, unlike the ordinary case, the variety of differential algebras with $L$-action generated by $M_k(F)$ has almost polynomial growth for all $k\geq 2$.

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arXiv2023

Differential identities and polynomial growth of the codimensions

Let $A$ be an associative algebra over a field $F$ of characteristic zero and let $L$ be a Lie algebra over $F$. If $L$ acts on $A$ by derivations, then such an action determines an action of its universal enveloping algebra $U(L)$ and in this case we refer to $A$ as algebra with derivations or $L$-algebra. Here we give a characterization of the ideal of differential identities of finite dimensional $L$-algebras $A$ in case the corresponding sequence of differential codimensions $c_n^L (A)$, $n\geq 1$, is polynomially bounded. As a consequence, we also characterize $L$-algebras with multiplicities of the differential cocharacter bounded by a constant.

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arXiv2022

Differential codimensions and exponential growth

Let $A$ be a finite dimensional associative algebra with derivations over a field of characteristic zero, i.e., an algebra whose structure is enriched by the action of a Lie algebra $L$ by derivations, and let $c_n^L(A),$ $n\geq 1,$ be its differential codimension sequence. Such sequence is exponentially bounded and $\exp^L(A) = \lim_{n\to \infty}\sqrt[n]{c_n^L(A)}$ is an integer that can be computed, called differential PI-exponent of $A$. In this paper we prove that for any Lie algebra $L$, $\exp^L(A)$ coincides with $\exp(A)$, the ordinary PI-exponent of $A$. Furthermore, in case $L$ is a solvable Lie algebra, we apply such result to classify varieties of $L$-algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth.

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arXiv2020

Growth of differential identities

In this paper we study the growth of the differential identities of some algebras with derivations, i.e., associative algebras where a Lie algebra $L$ (and its universal enveloping algebra $U(L)$) acts on them by derivations. In particular, we study in detail the differential identities and the cocharacter sequences of some algebras whose sequence of differential codimensions has polynomial growth. Moreover, we shall give a complete description of the differential identities of the algebra $UT_2$ of $2\times 2$ upper triangular matrices endowed with all possible action of a Lie algebra by derivations. Finally, we present the structure of the differential identities of the infinite dimensional Grassmann $G$ with respect to the action of a finite dimensional Lie algebra $L$ of inner derivations.

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