arXiv subjects

Plamen Koshlukov

Publications and source records attributed to Plamen Koshlukov.

At least 19 recordsLinked to original sources

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

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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arXiv2024

Gradings, graded identities, $$-identities and graded $$-identities of an algebra of upper triangular matrices

Let $K \langle X\rangle$ be the free associative algebra freely generated over the field $K$ by the countable set $X = \{x_1, x_2, \ldots\}$. If $A$ is an associative $K$-algebra, we say that a polynomial $f(x_1,\ldots, x_n) \in K \langle X\rangle$ is a polynomial identity, or simply an identity in $A$ if $f(a_1,\ldots, a_n) = 0$ for every $a_1, \ldots, a_n \in A$. Consider $\mathcal{A}$ the subalgebra of $UT_3(K)$ given by: \[ \mathcal{A} = K(e_{1,1} + e_{3,3}) \oplus Ke_{2,2} \oplus Ke_{2,3} \oplus Ke_{3,2} \oplus Ke_{1,3} , \] where $e_{i,j}$ denote the matrix units. We investigate the gradings on the algebra $\mathcal{A}$, determined by an abelian group, and prove that these gradings are elementary. Furthermore, we compute a basis for the $\mathbb{Z}_2$-graded identities of $\mathcal{A}$, and also for the $\mathbb{Z}_2$-graded identities with graded involution. Moreover, we describe the cocharacters of this algebra.

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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

Specht property for the graded identities of the pair $(M_2(D), sl_2(D))$

Let $D$ be a Noetherian infinite integral domain, denote by $M_2(D)$ and by $sl_2(D)$ the $2\times 2$ matrix algebra and the Lie algebra of the traceless matrices in $M_2(D)$, respectively. In this paper we study the natural grading by the cyclic group $\mathbb{Z}_2$ of order 2 on $M_2(D)$ and on $sl_2(D)$. We describe a finite basis of the graded polynomial identities for the pair $(M_2(D), sl_2(D))$. Moreover we prove that the ideal of the graded identities for this pair satisfies the Specht property, that is every ideal of graded identities of pairs (associative algebra, Lie algebra), satisfying the graded identities for $(M_2(D), sl_2(D))$, is finitely generated. The polynomial identities for $M_2(D)$ are known if $D$ is any field of characteristic different from 2. The identities for the Lie algebra $sl_2(D)$ are known when $D$ is an infinite field. The identities for the pair we consider were first described by Razmyslov when $D$ is a field of characteristic 0, and afterwards by the second author when $D$ is an infinite field. The graded identities for the pair $(M_2(D), gl_2(D))$ were also described, by Krasilnikov and the second author. In order to obtain these results we use certain graded analogues of the generic matrices, and also techniques developed by G. Higman concerning partially well ordered sets.

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arXiv2024

WEAK $G$-IDENTITIES FOR THE PAIR $(M_2( \mathbb{C}),sl_2( \mathbb{C}))$

In this paper we study algebras acted on by a finite group $G$ and the corresponding $G$-identities. Let $M_2( \mathbb{C})$ be the $2\times 2$ matrix algebra over the field of complex numbers $ \mathbb{C}$ and let $sl_2( \mathbb{C})$ be the Lie algebra of traceless matrices in $M_2( \mathbb{C})$. Assume that $G$ is a finite group acting as a group of automorphisms on $M_2( \mathbb{C})$. These groups were described in the Nineteenth century, they consist of the finite subgroups of $PGL_2( \mathbb{C})$, which are, up to conjugacy, the cyclic groups $ \mathbb{Z}_n$, the dihedral groups $D_n$ (of order $2n$), the alternating groups $ A_4$ and $A_5$, and the symmetric group $S_4$. The $G$-identities for $M_2( \mathbb{C})$ were described by Berele. The finite groups acting on $sl_2( \mathbb{C})$ are the same as those acting on $M_2( \mathbb{C})$. The $G$-identities for the Lie algebra of the traceless $sl_2( \mathbb{C})$ were obtained by Mortari and by the second author. We study the weak $G$-identities of the pair $(M_2( \mathbb{C}), sl_2( \mathbb{C}))$, when $G$ is a finite group. Since every automorphism of the pair is an automorphism for $M_2( \mathbb{C})$, it follows from this that $G$ is one of the groups above. In this paper we obtain bases of the weak $G$-identities for the pair $(M_2( \mathbb{C}), sl_2( \mathbb{C}))$ when $G$ is a finite group acting as a group of automorphisms.

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arXiv2023

On sums of gr-PI algebras

Let $A=B+C$ be an associative algebra graded by a group $G$, which is a sum of two homogeneous subalgebras $B$ and $C$. We prove that if $B$ is an ideal of $A$, and both $B$ and $C$ satisfy graded polynomial identities, then the same happens for the algebra $A$. We also introduce the notion of graded semi-identity for the algebra $A$ graded by a finite group and we give sufficient conditions on such semi-identities in order to obtain the existence of graded identities on $A$. We also provide an example where both subalgebras $B$ and $C$ satisfy graded identities while $A=B+C$ does not. Thus the theorem proved by Kȩpczyk in 2016 does not transfer to the case of group graded associative algebras. A variation of our example shows that a similar statement holds in the case of graded group Lie algebras. We note that there is no known analogue of Kȩpczyk's theorem for Lie algebras.

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arXiv2022

Specht property of varieties of graded Lie algebras

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.

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arXiv2022

Images of multilinear graded polynomials on upper triangular matrix algebras

In this paper we study the images of multilinear graded polynomials on the graded algebra of upper triangular matrices UT_n. For positive integers q \leq n, we classify these images on UT_n endowed with a particular elementary Z_q-grading. As a consequence, we obtain the images of multilinear graded polynomials on UT_n with the natural Z_n-grading. We apply this classification in order to give a new condition for a multilinear polynomial in terms of graded identities so that to obtain the traceless matrices in its image on the full matrix algebra. We also describe the images of multilinear polynomials on the graded algebras UT_2 and UT_3, for arbitrary gradings. We finish the paper by proving a similar result for the graded Jordan algebra UJ_2, and also for UJ_3 endowed with the natural elementary Z_3-grading.

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arXiv2021

$\mathbb{Z}$-graded identities of the Lie algebras $U_1$ in characteristic 2

Let $K$ be any field of characteristic two and let $U_1$ and $W_1$ be the Lie algebras of the derivations of the algebra of Laurent polynomials $K[t,t^{-1}]$ and of the polynomial ring $K[t]$, respectively. The algebras $U_1$ and $W_1$ are equipped with natural $\mathbb{Z}$-gradings. In this paper, we provide bases for the graded identities of $U_1$ and $W_1$, and we prove that they do not admit any finite basis.

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arXiv2021

A note on $\mathbb{Z}$-gradings on the Grassmann algebra and Elementary Number Theory

Let $E$ be the Grassmann algebra of an infinite dimensional vector space $L$ over a field of characteristic zero. In this paper, we study the $\mathbb{Z}$-gradings on $E$ having the form $E=E_{(r_{1},r_{2}, r_{3})}^{(v_{1},v_{2}, v_{3})}$, in which each element of a basis of $L$ has $\mathbb{Z}$-degree $r_{1}$, $r_{2}$, or $r_{3}$. We provide a criterion for the support of this structure to coincide with a subgroup of the group $\mathbb{Z}$, and we describe the graded identities for the corresponding gradings. We strongly use Elementary Number Theory as a tool, providing an interesting connection between this classical part of Mathematics, and PI Theory. Our results are generalizations of the approach presented in [11].

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arXiv2021

$\mathbb{Z}$-graded identities of the Lie algebras $U_1$

Let $K$ be an infinite field of characteristic different from two and let $U_1$ be the Lie algebra of the derivations of the algebra of Laurent polynomials $K[t,t^{-1}]$. The algebra $U_1$ admits a natural $\mathbb{Z}$-grading. We provide a basis for the graded identities of $U_1$ and prove that they do not admit any finite basis. Moreover, we provide a basis for the identities of certain graded Lie algebras with a grading such that every homogeneous component has dimension $\leq 1$, if a basis of the multilinear graded identities is known. As a consequence of this latter result we are able to provide a basis of the graded identities of the Lie algebra $W_1$ of the derivations of the polynomial ring $K[t]$. The $\mathbb{Z}$-graded identities for $W_1$, in characteristic 0, were described in \cite{FKK}. As a consequence of our results, we give an alternative proof of the main result, Theorem 1, in \cite{FKK}, and generalize it to positive characteristic. We also describe a basis of the graded identities for the special linear Lie algebra $sl_q(K)$ with the Pauli gradings where $q$ is a prime number.

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arXiv2021

Gradings on the algebra of triangular matrices as a Lie algebra: revisited

We investigate the group gradings on the algebra of upper triangular matrices over an arbitrary field, viewed as a Lie algebra. These results were obtained a few years early by the same authors. We provide streamlined proofs, and present a complete classification of isomorphism classes of the gradings. We also provide a classification of the practical isomorphism classes of the gradings, which is a better alternative way to consider these gradings up to being essentially the same object. Finally, we investigate in details the case where the characteristic of the base field is $2$, a topic that was neglected in previous works.

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arXiv2020

Matrix algebras with degenerate traces and trace identities

In this paper we study matrix algebras with a degenerate trace in the framework of the theory of polynomial identities. The first part is devoted to the study of the algebra $D_n$ of $n \times n$ diagonal matrices. We prove that, in case of a degenerate trace, all its trace identities follow by the commutativity law and by pure trace identities. Moreover we relate the trace identities of $D_{n+1}$ endowed with a degenerate trace, to those of $D_n$ with the corresponding trace. This allows us to determine the generators of the trace T-ideal of $D_3$. In the second part we study commutative subalgebras of $M_k(F)$, denoted by $C_k$ of the type $F + J$ that can be endowed with the so-called strange traces: $tr(a+j) = αa + βj$, for any $a+j \in C_k$, $α$, $β\in F$. Here $J$ is the radical of $C_k$. In case $β= 0$ such a trace is degenerate, and we study the trace identities satisfied by the algebra $C_k$, for every $k \geq 2$. Moreover we prove that these algebras generate the so-called minimal varieties of polynomial growth. In the last part of the paper, devoted to the study of varieties of polynomial growth, we completely classify the subvarieties of the varieties of algebras of almost polynomial growth introduced in an earlier paper of the same authors.

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arXiv2020

Trace identities and almost polynomial growth

In this paper we study algebras with trace and their trace polynomial identities over a field of characteristic 0. We consider two commutative matrix algebras: $D_2$, the algebra of $2\times 2$ diagonal matrices and $C_2$, the algebra of $2 \times 2$ matrices generated by $e_{11}+e_{22}$ and $e_{12}$. We describe all possible traces on these algebras and we study the corresponding trace codimensions. Moreover we characterize the varieties with trace of polynomial growth generated by a finite dimensional algebra. As a consequence, we see that the growth of a variety with trace is either polynomial or exponential.

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arXiv2020

2-graded polynomial identities for the Jordan algebra of the symmetric matrices of order two

The Jordan algebra of the symmetric matrices of order two over a field $K$ has two natural gradings by $\mathbb{Z}_2$, the cyclic group of order 2. We describe the graded polynomial identities for these two gradings when the base field is infinite and of characteristic different from 2. We exhibit bases for these identities in each of the two cases. In one of the cases we perform a series of computations in order to reduce the problem to dealing with associators while in the other case one employs methods and results from Invariant theory. Moreover we extend the latter grading to a $\mathbb{Z}_2$-grading on $B_n$, the Jordan algebra of a symmetric bilinear form in a vector space of dimension $n$ ($n=1$, 2, \dots, $\infty$). We call this grading the \textsl{scalar} one since its even part consists only of the scalars. As a by-product we obtain finite bases of the $\mathbb{Z}_2$-graded identities for $B_n$. In fact the last result describes the weak Jordan polynomial identities for the pair $(B_n, V_n)$.

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arXiv2020

Automorphisms and superalgebra structures on the Grassmann algebra

Let $F$ be a field of characteristic zero and let $E$ be the Grassmann algebra of an infinite dimensional $F$-vector space $L$. In this paper we study the superalgebra structures (that is the $\mathbb{Z}_{2}$-gradings) that the algebra $E$ admits. By using the duality between superalgebras and automorphisms of order $2$ we prove that in many cases the $\mathbb{Z}_{2}$-graded polynomial identities for such structures coincide with the $\mathbb{Z}_{2}$-graded polynomial identities of the "typical" cases $E_{\infty}$, $E_{k^\ast}$ and $E_{k}$ where the vector space $L$ is homogeneous. Recall that these cases were completely described by Di Vincenzo and Da Silva in \cite{disil}. Moreover we exhibit a wide range of non-homogeneous $\mathbb{Z}_{2}$-gradings on $E$ that are $\mathbb{Z}_{2}$-isomorphic to $E_{\infty}$, $E_{k^\ast}$ and $E_{k}$. In particular we construct a $\mathbb{Z}_{2}$-grading on $E$ with only one homogeneous generator in $L$ which is $\mathbb{Z}_{2}$-isomorphic to the natural $\mathbb{Z}_{2}$-grading on $E$, here denoted by $E_{can}$.

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