# 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

- email: e.hristova@math.bas.bg
- web: http://math.bas.bg
- phone: +3592 979 2824

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences

Ruhr University Bochum, Germany

PhD Thesis: Branching laws for tensor modules over classical locally finite Lie algebras

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

Bachelor's degree, Computer Science

Vesselin Drensky, Elitza Hristova

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

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

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

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],
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.

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