# Dimitar GrantcharovUniversity of Texas at Arlington | UTA · Department of Mathematics

Dimitar Grantcharov

Ph.D.

## About

55

Publications

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564

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Citations since 2016

Introduction

Dimitar Grantcharov currently works at the Department of Mathematics, University of Texas at Arlington. Dimitar does research in Lie algebras, representation theory, and geometry.

## Publications

Publications (55)

Let 𝒲 n {{\mathcal{W}}_{n}} be the Lie algebra of polynomial vector fields. We classify simple weight 𝒲 n {{\mathcal{W}}_{n}} -modules M with finite weight multiplicities. We prove that every such nontrivial module M is either a tensor module or the unique simple submodule in a tensor module associated with the de Rham complex on ℂ n {\mathbb{C}^{n...

We classify simple bounded weight modules over the complex simple Lie superalgebras $\mathfrak{sl}(\infty |\infty)$ and $\mathfrak{osp} (m | 2n)$, when at least one of $m$ and $n$ equals $\infty$. For $\mathfrak{osp} (m | 2n)$ such modules are of spinor-oscillator type, i.e., they combine into one the known classes of spinor $\mathfrak{o} (m)$-modu...

With the aid of the exponentiation functor and Fourier transform we introduce a class of modules T(g,V,S) of sl(n+1) of mixed tensor type. By varying the polynomial g, the gl(n)-module V, and the set S, we obtain important classes of weight modules over the Cartan subalgebra h of sl(n+1), and modules that are free over h. Furthermore, these modules...

We classify all simple bounded highest weight modules of a basic classical Lie superalgebra g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{g} $$\end{docume...

We introduce a new quantized enveloping superalgebra \({\mathfrak {U}}_q{\mathfrak {p}}_n\) attached to the Lie superalgebra \({\mathfrak {p}}_n\) of type P. The superalgebra \({\mathfrak {U}}_q{\mathfrak {p}}_n\) is a quantization of a Lie bisuperalgebra structure on \({\mathfrak {p}}_n\), and we study some of its basic properties. We also introdu...

We provide a classification and an explicit realization of all simple Gelfand–Tsetlin modules of the complex Lie algebra [Formula: see text]. The realization of these modules, including those with infinite-dimensional weight spaces, is given via regular and derivative Gelfand–Tsetlin tableaux. Also, we show that all simple Gelfand–Tsetlin [Formula:...

In this paper the authors introduce a class of parabolic subalgebras for classical simple Lie superalgebras associated to the detecting subalgebras introduced by Boe, Kujawa and Nakano. These parabolic subalgebras are shown to have good cohomological properties governed by the Bott-Borel-Weil theorem involving the zero component of the Lie superalg...

Let ${\mathcal W}_n$ be the Lie algebra of polynomial vector fields. We classify simple weight ${\mathcal W}_n$-modules $M$ with finite weight multiplicities. We prove that every such nontrivial module $M$ is either a tensor module or the unique simple submodule in a tensor module associated with the de Rham complex on $\mathbb C^n$.

In this paper we study properties of a homomorphism $\rho$ from the universal enveloping algebra $U=U(\mathfrak{gl}(n+1))$ to a tensor product of an algebra $\mathcal D'(n)$ of differential operators and $U(\mathfrak{gl}(n))$. We find a formula for the image of the Capelli determinant of $\mathfrak{gl}(n+1)$ under $\rho$, and, in particular, of the...

We indicate a geometric relation between Laplace-Beltrami spectra and eigenfunctions on compact Riemannian symmetric spaces and the Borel-Weil theory using ideas from symplectic geometry and geometric quantization. This is done by associating via Marsden-Weinstein reduction a generalized flag manifold parametrizing all maximal totally geodesic tori...

We introduce a new quantized enveloping superalgebra $\mathfrak{U}_q{\mathfrak{p}}_n$ attached to the Lie superalgebra ${\mathfrak{p}}_n$ of type $P$. The superalgebra $\mathfrak{U}_q{\mathfrak{p}}_n$ is a quantization of a Lie bisuperalgebra structure on ${\mathfrak{p}}_n$ and we study some of its basic properties. We also introduce the periplecti...

With the aid of the exponentiation functor and Fourier transform we introduce a class of modules $T(g,V,S)$ of $\mathfrak{sl} (n+1)$ of mixed tensor type. By varying the polynomial $g$, the $\mathfrak{gl}(n)$-module $V$, and the set $S$, we obtain important classes of weight modules over the Cartan subalgebra $\mathfrak h$ of $\mathfrak{sl} (n+1)$,...

We classify the simple bounded weight modules of the Lie algebras \( \mathfrak{sl}\left(\infty \right),\kern0.5em \mathfrak{o}\left(\infty \right) \) and \( \mathfrak{sp}\left(\infty \right) \), and compute their annihilators in \( U\left(\mathfrak{sl}\left(\infty \right)\right),\kern0.5em U\left(\mathfrak{o}\left(\infty \right)\right),\kern0.5em U...

We introduce the notion of essential support of a simple Gelfand-Tsetlin gln-module as an attempt to understand the character formula of such module. This support detects the weights having maximal possible Gelfand-Tsetlin multiplicities. Using combinatorial tools we describe the essential supports of the simple socles of the universal tableaux mod...

We classify all simple bounded highest weight modules of a basic classical Lie superalgebra $\mathfrak g$. In particular, our classification leads to the classification of the simple weight modules with finite weight multiplicities over all classical Lie superalgebras. We also obtain some character formulas of strongly typical bounded highest weigh...

We provide a classification and an explicit realization of all irreducible Gelfand-Tsetlin modules of the complex Lie algebra sl(3). The realization of these modules uses regular and derivative Gelfand-Tsetlin tableaux. In particular, we list all simple Gelfand-Tsetlin sl(3)-modules with infinite-dimensional weight spaces. Also, we express all simp...

We introduce the notion of essential support of a simple Gelfand-Tsetlin $\mathfrak{gl}_n$-module as an important tool towards understanding the character formula of such module. This support detects the weights in the module having maximal possible Gelfand-Tsetlin multiplicities. Using combinatorial tools we describe the essential supports of the...

In this paper the authors introduce a class of parabolic subalgebras for classical simple Lie superalgebras associated to the detecting subalgebras introduced by Boe, Kujawa and Nakano. These parabolic subalgebras are shown to have good cohomological properties governed by the Bott-Borel-Weil theorem involving the zero component of the Lie superalg...

We classify the simple bounded weight modules of ${\mathfrak{sl}(\infty})$, ${\mathfrak{o}(\infty)}$ and ${\mathfrak{sp}(\infty)}$, and compute their annihilators in $U({\mathfrak{sl}(\infty}))$, $U({\mathfrak{o}(\infty))}$, $U({\mathfrak{sp}(\infty))}$, respectively.

In the present paper we study Gelfand-Tsetlin modules defined in terms of BGG differential operators. The structure of these modules is described with the aid of the Postnikov-Stanley polynomials introduced in [PS09]. These polynomials are used to identify the action of the Gelfand-Tsetlin subalgebra on the BGG operators. We also provide explicit b...

We provide complete description of the simple modules in the principal block of the category of Gelfand-Tsetlin modules of sl(3). In addition, we prove that every such module is a subquotient of a localization of a highest weight module.

We prove a uniqueness theorem for irreducible non-critical Gelfand-Tsetlin modules. The uniqueness result leads to a complete classification of the irreducible Gelfand-Tsetlin modules with 1-singularity. An explicit construction of such modules was given in \cite{FGR2}. In particular, we show that the modules constructed in \cite{FGR2} exhaust all...

We study weight modules of the Lie algebra (Formula presented.) of vector fields on (Formula presented.). A classification of all simple weight modules of (Formula presented.) with a uniformly bounded set of weight multiplicities is provided. To achieve this classification, we introduce a new family of generalized tensor (Formula presented.)-module...

We provide a categorification of (Formula presented.)-crystals on the singular (Formula presented.)-category (Formula presented.). Our result extends the (Formula presented.)-crystal structure on (Formula presented.) induced from the work of Bernstein-Frenkel-Khovanov. Further properties of the (Formula presented.)-crystal (Formula presented.) are...

Singular Gelfand-Tsetlin modules of index 2 are modules whose tableaux bases may have singular pairs but no singular triples of entries on each row. In this paper we construct singular Gelfand-Tsetlin modules for arbitrary singular character of index 2. Explicit bases of derivative tableaux and the action of the generators of $\mathfrak{gl}(n)$ are...

We introduce a symmetric monoidal category of modules over the direct limit queer superalgebra \({\mathfrak{q}}(\infty )\). This category can be defined in two equivalent ways with the aid of the large annihilator condition. Tensor products of copies of the natural and the conatural representations are injective objects in this category. We obtain...

We classify blocks of categories of weight and generalized weight modules of
algebras of twisted differential operators on P^n. Necessary and sufficient
conditions for these blocks to be tame and proofs that some of the blocks are
Koszul are provided. We also establish equivalences of categories between these
blocks and categories of bounded and ge...

The Lie algebra of vector fields on $R^m$ acts naturally on the spaces of
differential operators between tensor field modules. Its projective subalgebra
is isomorphic to $sl_{m+1}$, and its affine subalgebra is a maximal parabolic
subalgebra of the projective subalgebra with Levi factor $gl_m$. We prove two
results. First, we realize all injective...

We provide a classification and explicit bases of tableaux of all irreducible
generic Gelfand-Tsetlin modules for the Lie algebra gl(n).

The classical Gelfand-Tsetlin formulas provide a basis in terms of tableaux
and an explicit action of the generators of $\mathfrak{gl} (n)$ for every
irreducible finite-dimensional $\mathfrak{gl} (n)$-module. These formulas can
be used to define a $\mathfrak{gl} (n)$-module structure on some
infinite-dimensional modules - the so-called generic Gelf...

We provide a categorification of $\mathfrak{q}(2)$-crystals on the singular
$\mathfrak{gl}_{n}$-category ${\mathcal O}_{n}$. Our result extends the
$\mathfrak{gl}_{2}$-crystal structure on ${\rm Irr} ({\mathcal O}_{n})$ defined
by Bernstein-Frenkel-Khovanov. Further properties of the ${\mathfrak
q}(2)$-crystal ${\rm Irr}({\mathcal O}_{n})$ are also...

We use localization technique to construct new families of irreducible
modules of affine Kac-Moody algebras. In particular, localization is applied to
the first free field realization of the affine Lie algebra A_1^{(1)} or,
equivalently, to imaginary Verma modules.

We discuss how the twisted localization functor leads to a classification of the simple objects and a description of the injectives in various categories of weight modules. The article is a survey on existing results for finite-dimensional simple Lie algebras and superalgebras, affine Lie algebras, and algebras of differential operators.

The purpose of this paper is to collect some recent results on the representation theory of Lie superalgebras of type Q. Results on the centres, simple weight modules and crystal bases of these superalgebras are included.

A classification of the simple highest weight bounded modules of the queer
Lie superalgebras q(n) is obtained. To achieve this classification we introduce
a new combinatorial tool - the star action. Our result leads in particular to a
classification of all simple weight q(n)-modules.

We classify simple weight modules over infinite dimensional Weyl algebras and
realize them using the action on certain localizations of the polynomial ring.
We describe indecomposable projective and injective weight modules and deduce
from this a description of blocks of the category of weight modules by quivers
and relations. As a corollary we est...

We give an explicit classification of the cominuscule parabolic subalgebras
of all complex simple finite dimensional Lie superalgebras.

In this paper, we develop the crystal basis theory for the quantum queer
superalgebra $U_q(\mathfrak q(n))$. We define the notion of crystal bases and
prove the tensor product rule for $U_q(\mathfrak q(n))$-modules in the category
$O_int^{\geq 0}$. Our main theorem shows that every $U_q(\mathfrak
q(n))$-module in the category $O_int^{\geq 0}$ has a...

In this paper, we give an explicit combinatorial realization of the crystal B ( λ ) B(\lambda ) for an irreducible highest weight U q ( q ( n ) ) U_q(\mathfrak {q}(n)) -module V ( λ ) V(\lambda ) in terms of semistandard decomposition tableaux. We present an insertion scheme for semistandard decomposition tableaux and give algorithms for decomposin...

In this paper, we develop the crystal basis theory for the quantum queer
superalgebra $\Uq$. We define the notion of crystal bases, describe the tensor
product rule, and present the existence and uniqueness of crystal bases for
finite-dimensional $\Uq$-modules in the category $\mathcal{O}_{int}^{\ge 0}$.

In this paper, we investigate the structure of highest weight modules over the quantum queer superalgebra \({U_q(\mathfrak {q}(n))}\). The key ingredients are the triangular decomposition of \({U_q(\mathfrak {q}(n))}\) and the classification of finite dimensional irreducible modules over quantum Clifford superalgebras. The main results we prove are...

All simple weight modules with finite dimensional weight spaces over affine Lie algebras are classified. Comment: Preliminary version, 25 pages

In this paper, we investigate the structure of highest weight modules over the quantum queer superalgebra Uq(q(n)). The key ingredients are the triangular decomposition of Uq(q(n)) and the classification of finite dimensional irreducible modules over quantum Clifford superalgebras. The main results we prove are the classical limit theorem and the

We compare two combinatorial definitions of parabolic sets of roots. We show that these definitions are equivalent for simple finite dimensional Lie algebras, affine Lie algebras, and toroidal Lie algebras. In contrast, these definitions are not always equivalent for simple finite dimensional Lie superalgebras.

We give a description of the algebraic group Aut(g) of automorphisms of a simple finite-dimensional Lie superalgebra g over an algebraically closed field k of characteristic 0, which is obtained by viewing g as a module over a Levi subalgebra of its even part. As an application, we give a detailed description of the group of automorphism of the k-L...

In this paper we study the subcategory of cuspidal modules of the category of weight modules over the Lie algebra sl(n+1). Our main result is a complete classification and explicit description of the indecomposable cuspidal modules. Comment: 31 pages, corrections added, to appear in Adv. Math

Let $g$ be a finite dimensional simple Lie algebra. Denote by $\mathcal B$ the category of all bounded weight $g$-modules, i.e. those which are direct sum of their weight spaces and have uniformly bounded weight multiplicities. A result of Fernando shows that infinite-dimensional bounded weight modules exist only for $g=sl(n)$ and $g=sp(2n)$. If $g...

Let $\mathfrak g$ be a classical Lie superalgebra of type I or a Cartan-type Lie superalgebra {\bf W}$(n)$. We study weight $\mathfrak g$-modules using a method inspired by Mathieu's classification of the simple weight modules with finite weight multiplicities over reductive Lie algebras, \cite{M}. Our approach is based on the fact that every simpl...

We describe the structure of the algebraic group of automorphisms of all simple finite dimensional Lie superalgebras. Using this and \'etale cohomology considerations, we list all different isomorphism classes of the corresponding twisted loop superalgebras.

We find sufficient conditions for a principal toric bundle over compact K\"ahler manifolds to admit Calabi-Yau connections with torsion. With the aids of a topological classification, we construct such geometry on $n(S^2\times S^4)#(n+1)(S^3\times S^3)$

Let g be a classical Lie superalgebra of type I. We introduce coherent families of weight g-modules with bounded weight multiplicities, and establish a correspondence between cuspidal and highest weight submodules of these families by extending Mathieu's work [Ann. Inst. Fourier 50 (2000) 537]. This enables us to reduce the description of the g0-mo...