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
Elitza Hristova
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 $\GL(n, K)$-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 $\lambda$ such that the irreducible $\GL(n, K)$-module $V_{\lambda}$ appears in the decomposition of $K\left\langle X \right\rangle / I_{p+1}$ as a $\GL(n, K)$-module. As an application, we take $K = \CC$ and we consider the algebra of invariants $(\CC\left\langle X \right\rangle / I_{p+1})^G$ for $G = \SL(n, \CC)$, $\OO(n, \CC)$, $\SO(n, \CC)$, or $\Sp(2s, \CC)$ (for $n=2s$). By a theorem of Domokos and Drensky, $(\CC\left\langle X \right\rangle / I_{p+1})^G$ is finitely generated. We give an upper bound on the degree of generators of $(\CC\left\langle X \right\rangle / I_{p+1})^G$ in a minimal generating set. In a similar way, we consider also the algebra of invariants $(\CC\left\langle X \right\rangle / I_{p+1})^{G}$, where $G=\mathrm{UT}(n, \CC)$, and give an upper bound on the degree of generators in a minimal generating set. These results provide useful information about the invariants in $\CC\left\langle X \right\rangle^G$ 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 $\CC\left\langle X \right\rangle$ belongs to $I_p$.
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 finite dimensional 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 $S(W)^G$. Our method is based on simple algebraic computations and can be easily realized using popular software packages. Then we give many explicit examples for computing $H(S(W)^G, t)$. As an application, we consider the question of regularity of the algebra $S(W)^{\OO(n)}$. For $n=2$ and $n=3$ we give a complete list of modules $W$, so that if $S(W)^{\OO(n)}$ is regular then $W$ is in this list. 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
In this paper, we consider the exterior algebra $\Lambda(W)$ of a polynomial $\GL(n)$-module $W$ and use previously developed methods to determine the Hilbert series of the algebra of invariants $\Lambda(W)^G$, where $G$ is one of the classical complex subgroups of $\GL(n)$, namely $\SL(n)$, $\OO(n)$, $\SO(n)$, or $\Sp(2d)$ (for $n=2d$). Since $\Lambda(W)^G$ is finite dimensional, we apply the described method to compute a lot of explicit examples. For $\Lambda(S^3\CC^3)^{\SL(3)}$, using the computed Hilbert series, we obtain an explicit set of generators.
Vesselin Drensky, Elitza Hristova
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.
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. 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 in 2009, our results hold for all possible embeddings $\g' \subset \g$ unless $\g' \cong \gl(\infty)$.
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