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Dimitar Grantcharov

Publications and source records attributed to Dimitar Grantcharov.

At least 37 records · Page 2Linked to original sources
arXiv2018

Gelfand-Tsetlin Theory for Rational Galois Algebras

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 bases of the corresponding Gelfand-Tsetlin modules and prove a simplicity criterion for these modules. The results hold for modules defined over standard Galois orders of type $A$ - a large class of rings that include the universal enveloping algebra of $\mathfrak{gl} (n)$ and the finite $W$-algebras of type $A$.

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arXiv2017

Quantization and injective submodules of differential operator modules

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 objects of the parabolic category O$^{gl_m}(sl_{m+1})$ of $gl_m$-finite $sl_{m+1}$-modules as submodules of differential operator modules. Second, we study projective quantizations of differential operator modules, i.e., $sl_{m+1}$-invariant splittings of their order filtrations. In the case of modules of differential operators from a tensor density module to an arbitrary tensor field module, we determine when there exists a unique projective quantization, when there exists no projective quantization, and when there exist multiple projective quantizations.

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arXiv2017

Drinfeld category and the classification of singular Gelfand-Tsetlin gl_n-modules

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 irreducible Gelfand-Tsetlin modules with 1-singularity. To prove the result we introduce a new category of modules (called Drinfeld category) related to the Drinfeld generators of the Yangian Y(gl_n) and define a functor from the category of non-critical Gelfand-Tsetlin modules to the Drinfeld category.

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arXiv2017

Bounded weight modules of the Lie algebra of vector fields on ${\mathbb C}^2$

We study weight modules of the Lie algebra $W_2$ of vector fields on ${\mathbb C}^2$. A classification of all simple weight modules of $W_2$ with a uniformly bounded set of weight multiplicities is provided. To achieve this classification we introduce a new family of generalized tensor $W_n$-modules. Our classification result is an important step in the classification of all simple weight $W_n$-modules with finite weight multiplicities.

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arXiv2017

New singular Gelfand-Tsetlin $\mathfrak{gl} (n)$-modules of index 2

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 given for these modules. Our construction leads to new families of irreducible Gelfand-Tsetlin modules and also provides tableaux bases for some simple Verma modules.

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arXiv2016

A categorification of $\mathfrak{q}(2)$-crystals

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 discussed.

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arXiv2016

Tensor representations of $\mathfrak q(\infty)$

We introduce a symmetric monoidal category of modules over the direct limit queer superalgebra $\q (\infty)$. The 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 the socle filtrations and formulas for the tensor products of the indecoposable injectives. In addition, it is proven that the category is Koszul self-dual.

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arXiv2015

Localization of free field realizations of affine Lie algebras

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.

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arXiv2014

Singular Gelfand-Tsetlin modules of $\mathfrak{gl}(n)$

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 Gelfand-Tsetlin modules. The generic Gelfand-Tsetlin modules are convenient to work with since for every generic tableau there exists a unique irreducible generic Gelfand-Tsetlin module containing this tableau as a basis element. In this paper we initiate the systematic study of a large class of non-generic Gelfand-Tsetlin modules - the class of $1$-singular Gelfand-Tsetlin modules. An explicit tableaux realization and the action of $\mathfrak{gl} (n)$ on these modules is provided using a new construction which we call derivative tableaux. Our construction of $1$-singular modules provides a large family of new irreducible Gelfand-Tsetlin modules of $\mathfrak{gl} (n)$, and is a part of the classification of all such irreducible modules for $n=3$.

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arXiv2013

On weight modules of algebras of twisted differential operators on the projective space

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 generalized bounded weight sl(n+1)-modules in the cases of nonintegral and singular central character.

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arXiv2013

Crystal bases for the quantum queer superalgebra and semistandard decomposition tableaux

In this paper, we give an explicit combinatorial realization of the crystal B(λ) for an irreducible highest weight U_q(q(n))-module V(λ) in terms of semistandard decomposition tableaux. We present an insertion scheme for semistandard decomposition tableaux and give algorithms of decomposing the tensor product of q(n)-crystals. Consequently, we obtain explicit combinatorial descriptions of the shifted Littlewood-Richardson coefficients.

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arXiv2013

Crystal bases for the quantum queer superalgebra

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 unique crystal basis.

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arXiv2013

Bounded highest weight modules of q(n)

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.

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arXiv2012

Weight modules over infinite dimensional Weyl algebras

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 establish Koszulity for all blocks.

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arXiv2011

Quantum Queer Superalgebra and Crystal Bases

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 $\mathcal{O}_{int}^{\ge 0}$.

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arXiv2011

Highest weight modules over quantum queer Lie superalgebra U_q(q(n))

In this paper, we investigate the structure of highest weight modules over the quantum queer superalgebra $U_q(q(n))$. The key ingredients are the triangular decomposition of $U_q(q(n))$ and the classification of finite dimensional irreducible modules over quantum Clifford superalgebras. The main results we prove are the classical limit theorem and the complete reducibility theorem for $U_q(q(n))$-modules in the category $O_q^{\geq 0}$.

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