\( \newcommand{\GL} {\mathrm{GL}} \newcommand{\SL} {\mathrm{SL}} \newcommand{\OO} {\mathrm{O}} \newcommand{\SO} {\mathrm{SO}} \newcommand{\Sp} {\mathrm{Sp}} \newcommand{\g} {\mathfrak{g}} \def\CC {{\mathbb C}} %% complex numbers \def\RR {{\mathbb R}} %% real numbers \def\NN {{\mathbb N}} %% natural numbers \def\QQ {{\mathbb Q}} %% rational numbers \def\ZZ {{\mathbb Z}} %% integers \def\PP {{\mathbb P}} %% projective \def\XX {{\mathbb X}} %% sphere \def\mk {\mathfrak} \newcommand{\gl} {\mathrm{gl}} \renewcommand{\sl} {\mathrm{sl}} \newcommand{\so} {\mathrm{so}} \renewcommand{\o} {\mathrm{o}} \renewcommand{\sp} {\mathrm{sp}} \)
Profile Picture of Elitza Hristova

Elitza Hristova

Assistant Professor, PhD

Research Positions

Sep 2015 - Present, Assistant Professor

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences

Mar 2014 - Jul 2015, Postdoctoral Researcher

Ruhr University Bochum, Germany

Education

2004 - 2007, Sofia University St. Kliment Ohridski

Master's degree, Theoretical and Mathematical Physics

Title of Master's Thesis: Geometric quantzation of $\mathbb{HP}^n$

2000 - 2004, Sofia University St. Kliment Ohridski

Bachelor's degree, Computer Science

Scientific Interests

Representation theory of Lie algebras and Lie groups; Applications to Invariant theory; Infinite-dimensional Lie algebras; Momentum maps; Geometric quantization and applications to Physics.

 

Publications

Vesselin Drensky, Elitza Hristova

Noncommutative invariant theory of symplectic and orthogonal groups, Linear Algebra and its Applications 581 (2019), 198-213. arXiv:1902.04164.

Abstract

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.


Vesselin Drensky, Elitza Hristova

Invariants of symplectic and orthogonal groups acting on $\GL(n,\CC)$-modules, submitted. arXiv:1707.05893.

Abstract

Let $\GL(n) = \GL(n, \CC)$ denote the complex general linear group and let $G \subset \GL(n)$ be one of the classical complex subgroups $\OO(n)$, $\SO(n)$, and $\Sp(2k)$ (in the case $n = 2k$). We take a polynomial $\GL(n)$-module $W$ and consider the symmetric algebra $S(W)$. Extending previous results for $G=\SL(n)$, 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 $\Lambda(S^2 V)^G$ and $\Lambda(\Lambda^2 V)^G$, where $V = \CC^n$ denotes the standard $\GL(n)$-module.


Elitza Hristova, Ivan Penkov

Decomposition of cohomology of vector bundles on homogeneous ind-spaces, C. R. Acad. Bulg. Sci. 70 (2017), No 7, 907-916. arXiv:1703.05086.

Abstract

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 $\OO_{G/P}(E^*)$ on $G/P$ induced by the dual $P$-module $E^*$ decompose as direct sums of cohomologies of bundles of the form $\OO_{G/P}(R)$ 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 $\OO_{G/P}(R)$.


Elitza Hristova, Tomasz Maciazek, Valdemar V. Tsanov

On momentum images of representations and secant varieties, preprint. arXiv:1504.01110.

Abstract

Let $K$ be a connected compact semisimple group and $V_\lambda$ be an irreducible unitary representation with highest weight $\lambda$. We study the momentum map $\mu:\PP(V_\lambda)\to\mk k^*$. The intersection $\mu(\PP)^+=\mu(\PP)\cap{\mk t}^+$ of the momentum image with a fixed Weyl chamber is a convex polytope called the momentum polytope of $V_\lambda$. We construct an affine rational polyhedral convex cone $\Upsilon_\lambda$ with vertex $\lambda$, such that $\mu(\PP)^+\subset\Upsilon_\lambda \cap {\mk t}^+$. 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 $\lambda$. We also present some results on the critical points of $||\mu||^2$. 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 $\XX\subset\PP$, and prove a relation between the momentum images of the secant varieties and the degrees of $K$-invariant polynomials on $V_\lambda$.


Elitza Hristova

Branching laws for tensor modules over classical locally finite Lie algebras, Journal of Algebra 397 (2014), 278-314. arXiv:1304.7937.

Abstract

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 $\g' \subset \g$ of general tensor type and derive branching laws for triples $\g', \g, M$, where $\g' \subset \g$ is an embedding of general tensor type and $M$ is a simple tensor $\g$-module. 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 in [DP], when $\g' \not\cong \gl(\infty)$, our results hold for all possible embeddings $\g' \subset \g$ unless $\g' \cong \gl(\infty)$.

[DP] I. Dimitrov, I. Penkov, Locally semisimple and maximal subalgebras of the finitary Lie algebras gl($\infty$), sl($\infty$), so($\infty$), and sp($\infty$), J. Algebra 322 (2009), 2069-2081.

Contacts

Department of Algebra and Logic

Institute of Mathematics and Informatics

Bulgarian Academy of Sciences

1113, Sofia, Bulgaria, Acad. Georgi Bonchev Str., Block 8, Room 562

email: e.hristova@math.bas.bg

phone: +3592 979 2824