Searcharxiv⌕ Search

arXiv subjects

Vyacheslav Futorny

Publications and source records attributed to Vyacheslav Futorny.

At least 127 records · Page 7Linked to original sources
arXiv2010

Torsion theories induced from commutative subalgebras

We begin a study of torsion theories for representations of an important class of associative algebras over a field which includes all finite W-algebras of type A, in particular the universal enveloping algebra of gl(n) (or sl(n)) for all n. If U is such and algebra which contains a finitely generated commutative subalgebra A, then we show that any A-torsion theory defined by the coheight of prime ideals is liftable to U. Moreover, for any simple U-module M, all associated prime ideals of M in Spec A have the same coheight. Hence,thecoheight of the associated prime ideals of A is an invariant of a given simple U-module. This implies a stratification of the category of $U$-modules controlled by the coheight of associated prime ideals of A. Our approach can be viewed as a generalization of the classical paper by R.Block, it allows in particular to study representations of gl(n) beyond the classical category of weight or generalized weight modules.

math.RT↗
arXiv2009

Fibers of characters in Harish-Chandra categories

We solve the problem of extension of characters of commutative subalgebras in associative (noncommutative) algebras for a class of subrings (Galois orders) in skew group rings. These results can be viewed as a noncommutative analogue of liftings of prime ideals in the case of integral extensions of commutative rings. The proposed approach can be applied to the representation theory of many infinite dimensional algebras including universal enveloping algebras of reductive Lie algebras, Yangians and finite $W$-algebras. In particular, we develop a theory of Gelfand-Tsetlin modules for $\gl_n$. Besides classification results we characterize their categories in the generic case extending the classical results on $\gl_2$.

math.RT↗
arXiv2009

Gelfand-Kirillov Conjecture and Harish-Chandra Modules for Finite W-Algebras

We address two problems regarding the structure and representation theory of finite W-algebras associated with the general linear Lie algebras. Finite W-algebras can be defined either via the Whittaker model of Kostant or, equivalently, by the quantum Hamiltonian reduction. Our first main result is a proof of the Gelfand-Kirillov conjecture for the skew fields of fractions of the finite W-algebras. The second main result is a parametrization of finite families of irreducible Harish-Chandra modules by the characters of the Gelfand-Tsetlin subalgebra. As a corollary, we obtain a complete classification of generic irreducible Harish-Chandra modules for the finite W-algebras.

math.RA↗
arXiv2009

Free Field Realizations of the Elliptic Affine Lie Algebra sl(2,R)+Ω_R/dR

In this paper we construct two free field realizations of the elliptic affine Lie algebra sl(2,R) + Ω_R/dR, where R=C[t,t^{-1},u|u^2=t^3 - 2b t^2 + t]. The first realization gives an analogue of Wakimoto's construction for Affine Kac-Moody algebras, but in the setting of the elliptic affine Lie algebra. The second realization gives new type of representations analogous to Imaginary Verma modules in the Affine setting.

math.RT↗
arXiv2009

Parabolic sets of roots

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.

math.RT↗
arXiv2009

Induced Modules for Affine Lie Algebras

We study induced modules of nonzero central charge with arbitrary multiplicities over affine Lie algebras. For a given pseudo parabolic subalgebra ${\mathcal P}$ of an affine Lie algebra ${\mathfrak G}$, our main result establishes the equivalence between a certain category of ${\mathcal P}$-induced ${\mathfrak G}$-modules and the category of weight ${\mathcal P}$-modules with injective action of the central element of ${\mathfrak G}$. In particular, the induction functor preserves irreducible modules. If ${\mathcal P}$ is a parabolic subalgebra with a finite-dimensional Levi factor then it defines a unique pseudo parabolic subalgebra ${\mathcal P}^{ps}$, ${\mathcal P}\subset {\mathcal P}^{ps}$. The structure of ${\mathcal P}$-induced modules in this case is fully determined by the structure of ${\mathcal P}^{ps}$-induced modules. These results generalize similar reductions in particular cases previously considered by V. Futorny, S. König, V. Mazorchuk [Forum Math. 13 (2001), 641-661], B. Cox [Pacific J. Math. 165 (1994), 269-294] and I. Dimitrov, V. Futorny, I. Penkov [Comm. Math. Phys. 250 (2004), 47-63].

math.RT↗
arXiv2008

Tilting, deformations and representations of linear groups over Euclidean algebras

We consider the dual space of linear groups over Dynkinian and Euclidean algebras, i.e. finite dimensional algebras derived equivalent to the path algebra of Dynkin or Euclidean quiver. We prove that this space contains an open dense subset isomorphic to the product of dual spaces of full linear groups and, perhaps, one more (explicitly described) space. The proof uses the technique of bimodule categories, deformations and representations of quivers.

math.RT↗
arXiv2008

Galois orders

We introduce and develop a structure theory of a new class of noncommutative rings - Galois orders, that generalize classical orders in noncommutative rings. Galois orders realized as certain subrings of invariants in skew semigroup rings. This class of rings contains many classical objects, in particular the Generalized Weyl algebras, the universal enveloping algebra of the general linear Lie algebra and certain rings of invariant differential operators on algebraic varieties.

math.RT↗
arXiv2007

On Moduli Spaces for Abelian Categories

We show that if A is an abelian category satisfying certain mild conditions, then one can introduce the concept of a moduli space of (semi)stable objects which has the structure of a projective algebraic variety. This idea is applied to several important abelian categories in representation theory, like highest weight categories.

math.AG↗
arXiv2007

Integrable modules for affine Lie superalgebras

Irreducible nonzero level modules with finite-dimensional weight spaces are studied for non-twisted affine Lie superalgebras. A complete classification is obtained for superalgebras A(m,n)^ and C(n)^. In other cases the classification problem is reduced to the classification of cuspidal modules over finite-dimensional cuspidal Lie superalgebras described by Dimitrov, Mathieu and Penkov. It is also shown that any irreducible weakly integrable (in the sense of Kac and Wakimoto) module is highest weight.

math.RT↗
arXiv2006

Structure of Intermediate Wakimoto Modules

We show that our construction of boson type realizations of affine $\mathfrak{sl}(n+1)$ in terms of intermediate Wakimoto modules gives representations that are generically isomorphic to certain Verma type modules. We then use this identification to obtain information about the submodule structure of intermediate Wakimoto modules

math.RT↗
arXiv2003

Intermediate Wakimoto modules for Affine sl(n+1)

We construct certain boson type realizations of affine sl(n+1) that depend on a parameter r. When r=0 we get a Fock space realization of Imaginary Verma modules appearing in the work of the first author and when r=n they are the Wakimoto modules described in the work of Feigin and Frenkel.

math.RT↗