arXiv subjects

Elitza Hristova

Publications and source records attributed to Elitza Hristova.

12 recordsLinked to original sources

arXiv2026

On the cocharacter sequence of some PI-algebras

Let $A$ be a unital associative PI-algebra over a field of characteristic zero. We study which partitions $λ$ appear with nonzero multiplicities in the cocharacter sequence of $A$ for several classes of algebras $A$. Berele defines the eventual arm width $ω_0(A)$ to be the maximal integer $d$ so that if $λ$ appears with nonzero multiplicity in the cocharacter sequence of $A$, then $λ$ can have at most $d$ parts arbitrarily large. Berele also shows that if $A$ is Lie nilpotent, then $ω_0(A) = 1$. In the first part of this paper, we show that if $A$ is unital, then $ω_0(A) = 1$ if and only if $A$ is Lie nilpotent. To prove this statement, we show that the algebra of proper polynomials $B_n(A)$ is finite dimensional if and only if $A$ is Lie nilpotent. In the second part, we give a bound on the nonzero multiplicities $λ$ in the cocharacter sequence of $A$, when the T-ideal of identities of $A$ is equal to a product of T-ideals generated by long commutators. As an application, we show that for a Lie nilpotent algebra $A$, the nonzero multiplicities $m_λ(A)$ correspond to partitions $λ$ which are supported in step-like diagrams in which the number of steps grows with the index of Lie nilpotency. Finally, we give also some applications to the noncommutative invariant theory of the special linear group $\mathrm{SL}(n)$.

math.RA↗

arXiv2025

Tensor products of Lie nilpotent associative algebras and applications to codimension sequences

Let $G$ and $H$ be Lie nilpotent associative algebras over a field $K$ such that in addition $H$ satisfies the identity $[x_1, x_2] \cdots [x_{2k-1}, x_{2k}]=0$ for some $k \geq 2$. In this paper, extending results of Deryabina and Krasilnikov, we show that the tensor product $G \otimes H$ is again a Lie nilpotent associative algebra. Moreover, we give a lower and an upper bound on the minimal value of $q$ for which $[x_1, \dots, x_{q+1}] = 0$ is an identity for $G\otimes H$. In the case when $H$ satisfies the identities $[x_1, x_2, x_3] = 0$ and $[x_1, x_2][x_3, x_4] = 0$ and $\operatorname{char}K \neq 3$, we determine a better upper bound for $q$, which in many cases is equal to the minimal index of Lie nilpotency for $G\otimes H$. As a corollary, we reprove a result of Drensky saying that any product of Grassmann algebras of the form $E\otimes E_{i_1}\otimes \cdots \otimes E_{i_s}$ or $E_{j_1} \otimes E_{j_2} \otimes \cdots \otimes E_{j_t}$, where $E$ denotes the Grassmann algebra over a countable dimensional vector space and $E_r$ denotes the Grassmann algebra over an $r$-dimensional vector space, satisfies an identity of the form $[x_1, \dots, x_{q+1}] = 0$. We also provide several particular cases in which the minimal value of $q$ can be explicitly computed. As an application, we consider a field of characteristic zero, the variety $\mathfrak{N}_p$ of Lie nilpotent associative algebras of index at most $p$ and the corresponding relatively free algebras of finite rank, $F_n(\mathfrak{N}_p)$. We exhibit many explicit irreducible $S_n$-modules in the $S_n$-module decomposition of the space of proper multilinear polynomials of degree $n$ in $F_n(\mathfrak{N}_p)$ for any $p$. This gives a lower bound for the dimensions of the spaces of multilinear and proper multilinear polynomials of degree $n$ in $F_n(\mathfrak{N}_p)$.

math.RA↗

arXiv2025

Identities of relatively free algebras of Lie nilpotent associative algebras

In this paper, we consider the relatively free algebra of rank $n$, $F_n(\mathfrak{N}_p)$, in the variety of Lie nilpotent associative algebras of index $p$, denoted by $\mathfrak{N}_p$, over a field of characteristic zero. We describe an explicit minimal basis for the polynomial identities of $F_n(\mathfrak{N}_p)$ when $p=3$ and $p=4$, for all $n$, except for $F_3(\mathfrak{N}_4)$. In the general case, we exhibit a lower and an upper bound for the minimal $k$ such that $[x_1,x_2]\cdots[x_{2k-1},x_{2k}]$ is an identity for $F_n(\mathfrak{N}_p)$ for all $n$ and for all $p$.

math.RA↗

arXiv2025

A geometry of cubic discriminants in 8 dimensions

This paper examines 8-dimensional Riemannian manifolds whose structure group reduces to ${SO(4)}_{ir}\subset GL(8,\mathbb R)$, the image of an irreducible representation of $SO(4)$ on $\mathbb R^8$. We demonstrate that such a reduction can be described by an almost quaternion-Hermitian structure and a special rank-4 tensor field, which we call a cubic discriminant. This tensor field is pointwise linearly equivalent to the formula for the discriminant of a cubic polynomial. We show that the only non-flat, integrable examples of these structures are the quaternion-Kähler symmetric spaces $G_2\big/SO(4)$ and $G_{2(2)}\big/SO(4)$. We also present a new curvature-based characterization for the Riemannian metrics on these spaces.

math.DG↗

arXiv2024

Actions of frieze groups on inverse limits of polynomial rings

In the spirit of the action of the symmetric group on the ring of polynomials in $n$ variables, we consider the actions of the seven frieze groups on rings of formal infinite linear combinations of monomials of restricted degree. For each group we describe the respective subring of invariants. We discuss also the structure of those rings as modules over each frieze group.

math.GR↗

arXiv2022

On the $\mathrm{GL}(n)$-module structure of Lie nilpotent associative relatively free algebras

Let $K\left\langle X \right\rangle$ denote the free associative algebra generated by a set $X = \{x_1, \dots, x_n\}$ over a field $K$ of characteristic $0$. Let $I_p$, for $p \geq 2$, denote the two-sided ideal in $K\left\langle X \right\rangle$ generated by all commutators of the form $[u_1, \dots, u_p]$, where $u_1, \dots, u_p \in K\left\langle X \right\rangle$. We discuss the $\mathrm{GL}(n, K)$-module structure of the quotient $K\left\langle X \right\rangle / I_{p+1}$ for all $p \geq 1$ under the standard diagonal action. We give a bound on the values of partitions $λ$ such that the irreducible $\mathrm{GL}(n, K)$-module $V_λ$ appears in the decomposition of $K\left\langle X \right\rangle / I_{p+1}$ as a $\mathrm{GL}(n, K)$-module. As an application, we take $K = \mathbb{C}$ and we consider the algebra of invariants $(\mathbb{C}\left\langle X \right\rangle / I_{p+1})^G$ for $G = \mathrm{SL}(n, \mathbb{C})$, $\mathrm{O}(n, \mathbb{C})$, $\mathrm{SO}(n, \mathbb{C})$, or $\mathrm{Sp}(2s, \mathbb{C})$ (for $n=2s$). By a theorem of Domokos and Drensky, $(\mathbb{C}\left\langle X \right\rangle / I_{p+1})^G$ is finitely generated. We give an upper bound on the degree of generators of $(\mathbb{C}\left\langle X \right\rangle / I_{p+1})^G$ in a minimal generating set. In a similar way, we consider also the algebra of invariants $(\mathbb{C}\left\langle X \right\rangle / I_{p+1})^{G}$, where $G=\mathrm{UT}(n, \mathbb{C})$, and give an upper bound on the degree of generators in a minimal generating set. These results provide useful information about the invariants in $\mathbb{C}\left\langle X \right\rangle^G$ from the point of view of Classical Invariant Theory. In particular, for all $G$ as above we give a criterion when a $G$-invariant of $\mathbb{C}\left\langle X \right\rangle$ belongs to $I_p$.

math.RA↗

arXiv2020

Invariants of symplectic and orthogonal groups acting on $\text{GL}(n,{\mathbb C})$-modules

Let $\text{GL}(n) = \text{GL}(n, {\mathbb C})$ denote the complex general linear group and let $G \subset \text{GL}(n)$ be one of the classical complex subgroups $\text{O}(n)$, $\text{SO}(n)$, and $\text{Sp}(2k)$ (in the case $n = 2k$). We take a polynomial $\text{GL}(n)$-module $W$ and consider the symmetric algebra $S(W)$. Extending previous results for $G=\text{SL}(n)$, we develop a method for determining the Hilbert series $H(S(W)^G, t)$ of the algebra of invariants $S(W)^G$. Then we give explicit examples for computing $H(S(W)^G, t)$. As a further application, we extend our method to compute also the Hilbert series of the algebras of invariants $Λ(S^2 V)^G$ and $Λ(Λ^2 V)^G$, where $V = {\mathbb C}^n$ denotes the standard $GL(n)$-module.

math.AC↗

arXiv2019

Hilbert series and invariants in exterior algebras

In this paper, we consider the exterior algebra $Λ(W)$ of a polynomial $\mathrm{GL}(n)$-module $W$ and use previously developed methods to determine the Hilbert series of the algebra of invariants $Λ(W)^G$, where $G$ is one of the classical complex subgroups of $\mathrm{GL}(n)$, namely $\mathrm{SL}(n)$, $\mathrm{O}(n)$, $\mathrm{SO}(n)$, or $\mathrm{Sp}(2d)$ (for $n=2d$). Since $Λ(W)^G$ is finite dimensional, we apply the described method to compute a lot of explicit examples. For $Λ(S^3\mathbb{C}^3)^{\mathrm{SL}(3)}$, using the computed Hilbert series, we obtain an explicit set of generators.

math.RT↗

arXiv2019

Noncommutative invariant theory of symplectic and orthogonal groups

We present a method for computing the Hilbert series of the algebra of invariants of the complex symplectic and orthogonal groups acting on graded noncommutative algebras with homogeneous components which are polynomial modules of the general linear group. We apply our method to compute the Hilbert series for different actions of the symplectic and orthogonal groups on the relatively free algebras of the varieties of associative algebras generated, respectively, by the Grassmann algebra and the algebra of $2\times 2$ upper triangular matrices. These two varieties are remarkable with the property that they are the only minimal varieties of exponent 2.

math.RA↗

arXiv2017

Decomposition of cohomology of vector bundles on homogeneous ind-spaces

Let $G$ be a locally semisimple ind-group, $P$ be a parabolic subgroup, and $E$ be a finite-dimensional $P$-module. We show that, under a certain condition on $E$, the nonzero cohomologies of the homogeneous vector bundle $\mathcal{O}_{G/P}(E^*)$ on $G/P$ induced by the dual $P$-module $E^*$ decompose as direct sums of cohomologies of bundles of the form $\mathcal{O}_{G/P}(R)$ for (some) simple constituents $R$ of $E^*$. In the finite-dimensional case, this result is a consequence of the Bott-Borel-Weil theorem and Weyl's semisimplicity theorem. In the infinite-dimensional setting we consider, there is no relevant semisimplicity theorem. Instead, our results are based on the injectivity of the cohomologies of the bundles $\mathcal{O}_{G/P}(R)$.

math.RT↗

arXiv2015

On momentum images of representations and secant varieties

Let $K$ be a connected compact semisimple group and $V_λ$ be an irreducible unitary representation with highest weight $λ$. We study the momentum map $μ:\mathbb P(V_λ)\to\mathfrak k^*$. The intersection $μ(\mathbb P(V_λ))^+=μ(\mathbb P(V_λ))\cap{\mathfrak t}^+$ of the momentum image with a fixed Weyl chamber is a convex polytope called the momentum polytope of $V_λ$. We construct an affine rational polyhedral convex cone $Υ_λ$ with vertex $λ$, such that $μ(\mathbb P(V_λ))^+\subsetΥ_λ\cap {\mathfrak t}^+$. We show that equality holds for a class of representations, including those with regular highest weight. For those cases, we obtain a complete combinatorial description of the momentum polytope, in terms of $λ$. We also present some results on the critical points of $||μ||^2$. Namely, we consider the existence problem for critical points in the preimages of Kirwan's candidates for critical values. Also, we consider the secant varieties to the unique complex orbit $\mathbb X\subset\mathbb P(V_λ)$, and prove a relation between the momentum images of the secant varieties and the degrees of $K$-invariant polynomials on $V_λ$.

math.RT↗

arXiv2013

Branching laws for tensor modules over classical locally finite Lie algebras

Let g' and g be isomorphic to any two of the Lie algebras gl(infty), sl(infty), sp(infty), and so(infty). Let M be a simple tensor g-module. We introduce the notion of an embedding of g' into g of general tensor type and derive branching laws for triples g', g, and M, where the embedding of g' into g is of general tensor type. More precisely, since M is in general not semisimple as a g'-module, we determine the socle filtration of M over g'. Due to the description of embeddings of classical locally finite Lie algebras given by Dimitrov and Penkov, our results hold for all possible embeddings of g' into g unless g' is isomorphic to gl(infty).

math.RT↗