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We provide an explicit combinatorial realization of all simple and injective (hence, and projective) modules in the category of bounded $\mathfrak{sp}(2n)$-modules. This realization is defined via a natural tableaux correspondence between spinor-type modules of $\mathfrak{so}(2n)$ and oscillator-type modules of $\mathfrak{sp}(2n)$. In particular, we show that, in contrast with the $A$-type case, the generic and bounded $\mathfrak{sp}(2n)$-modules admit an analog of the Gelfand-Graev continuation from finite-dimensional representations., Comment: 24 pages

We introduce a certain quantum superalgebra in the Drinfeld realization and show that the quantum affine superalgebra of type $B$ is its homomorphic image (conjecturally isomorphic). We also define a braid group action on quantum affine superalgebra for any choice of parity., Comment: 27 pages

Smooth modules for affine Kac-Moody algebras have a prime importance for the quantum field theory as they correspond to the representations of the universal affine vertex algebras. But, very little is known about such modules beyond the category of positive energy representations. We construct a new class of smooth modules over affine Kac-Moody algebras. In a particular case, these modules are isomorphic to those induced from generalized Whittaker modules for Takiff Lie algebras. We establish the irreducibility criterion for constructed modules in the case of the affine sl(2) Lie algebra.

The closure of a Diximier sheet is the image of a generalized Grothendieck's simultaneous resolution. We show that the associated variety of simple affine vertex algebras is contained in the closure of the Diximier sheet when a chiralization of generalized Grothendieck's simultaneous resolution exists. This generalizes in a conceptual manner the results obtained by the first named author and Anne Moreau and, in particular, allows to extend them to the types A_2, C_n, E_6, E_7.

We construct two families of orthogonal polynomials associated with the universal central extensions of the superelliptic Lie algebras. These polynomials satisfy certain fourth order linear differential equations, and one of the families is a particular collection of associated ultraspherical polynomials. We show that the generating functions of the polynomials satisfy fourth order linear PDEs. Since these generating function can be represented by superelliptic integrals, we have examples of linear PDEs of fourth order with explicit solutions without complete integrability.

We explicitly construct families of simple modules for Lie algebras of rank $2$, on which certain commutative subalgebra acts diagonally and has a simple spectrum. In type $A$ these modules are well known generic Gelfand-Tsetlin modules and they can be viewed as such for other rank $2$ Lie algebras.

In this paper, we study the category of modules over the Smith algebra which are free of finite rank over the unital polynomial subalgebra generated by the Cartan element $h$ and obtain families of such simple modules of arbitrary rank. In the case of rank one we obtain a full description of the isomorphism classes, a simplicity criterion, and an algorithm to produce all composition series. We show that all such modules have finite length and describe the composition factors and their multiplicity., Comment: 16 pages

We consider reduced imaginary Verma modules for the untwisted quantum affine algebras $U_q(\hat{\g})$ and define a crystal-like base which we call imaginary crystal base using the Kashiwara algebra $\mathcal K_q$ constructed in earlier work by Ben Cox and two of the authors. We prove the existence of the imaginary crystal base for any object in a suitable category $\mc{O}^q_{red,im}$ containing the reduced imaginary Verma modules for $U_q(\hat{\g})$., Comment: 26 pages

For an arbitrary affine Lie algebra we study an analog of the category O for the natural Borel subalgebra and zero central charge. We show that such category is semisimple having the reduced imaginary Verma modules as its simple objects. This generalizes the result of Cox, Futorny, Misra in the case of affine sl(2).

In this paper we show variant of the spectral theorem using an algebraic Jordan-Schwinger map. The advantage of this approach is that we don't have restriction of normality on the class of operators we consider. On the other side, we have the restriction that the class of operators we consider should be of weighted Hilbert-Schmidt class.

We consider an algebra of even-order square tensors and introduce a stretching map which allows us to represent tensors as matrices. The stretching map could be understood as a generalized matricization. It conserves algebraic properties of the tensors. In the same time, we don't necessarily assume injectivity of the stretching map. Dropping the injectivity condition allows us to construct examples of stretching maps with additional symmetry properties. Furthermore, the noninjectivity leads to the averaging of the tensor and possibly could be used to compress the data., Comment: 15 pages

We use induction from parabolic subalgebras with infinite-dimensional Levi factor to construct new families of irreducible representations for arbitrary Affine Kac-Moody algebra. Our first construction defines a functor from the category of Whittaker modules over the Levi factor of a parabolic subalgebra to the category of modules over the Affine Lie algebra. The second functor sends tensor products of a module over the affine part of the Levi factor (in particular any weight module) and of a Whittaker module over the complement Heisenberg subalgebra to the Affine Lie algebra modules. Both functors preserves irreducibility when the central charge is nonzero.

We construct a new family of irreducible modules over any basic classical affine Kac-Moody Lie superalgebra which are induced from modules over the Heisenberg subalgebra. We also obtain irreducible deformations of these modules for the quantum affine superalgebra of type A.

We provide an explicit combinatorial description of highest weights of simple highest weight modules over the simple affine vertex algebra of type A of admissible level k. For admissible simple highest weight modules corresponding to the principal, subregular and maximal parabolic nilpotent orbits we give a realization using the Gelfand-Tsetlin theory, which also allows us to obtain a realization of certain classes of simple admissible sl(2)-induced modules in these orbits. In particular, simple admissible sl(2)-induced modules are fully classified for the principal nilpotent orbit., Comment: 37 pages

In this note we define one more way of quantization of classical systems. The quantization we consider is an analogue of classical Jordan-Schwinger (J.-S.) map which has been known and used for a long time by physicists. The difference, comparing to J.-S. map, is that we use generators of Cuntz algebra $\mathcal{O}_{\infty}$ (i.e. countable family of mutually orthogonal partial isometries of separable Hilbert space) as a "building blocks" instead of creation-annihilation operators. The resulting scheme satisfies properties similar to Van Hove prequantization i.e. exact conservation of Lie bracket and linearity.

We obtain a classification of simple modules with finite weight multiplicities over basic classical map superalgebras. Any such module is parabolic induced from a simple cuspidal bounded module over a cuspidal map superalgebra. Further on, any simple cuspidal bounded module is isomorphic to an evaluation module. As an application, we obtain a classification of all simple Harish-Chandra modules for basic classical loop superalgebras. Extending these results to affine Kac-Moody Lie superalgebras obtained by adding the degree derivation, we construct a family of bounded simple modules of level zero, and conjecture that all bounded simple cuspidal modules belong to this family. Finally, we show that for affine Kac-Moody Lie superalgebras of type I the Kac induction functor reduces the classification of all simple bounded modules to the classification of the same class of modules over the even part, whose classification is claimed by Dimitrov and Grantcharov.

I.M. Gelfand and V.A. Ponomarev (1969) proved that the problem of classifying pairs (A,B) of commuting nilpotent operators on a vector space contains the problem of classifying an arbitrary t-tuple of linear operators. Moreover, it contains the problem of classifying representations of an arbitrary quiver, and so it is considered as hopeless. We give a simple normal form of the matrices of (A,B) if the intersection of kernels of A and B is one-dimensional. We prove that this form is canonical if the Jordan matrix of A is a direct sum of Jordan blocks of the same size and the field is of zero characteristic.

We study holonomic modules for the rings of invariant differential operators on affine commutative domains with finite Krull dimension with respect to arbitrary actions of finite groups. We prove the Bernstein inequality for these rings. Our main tool is the filter dimension introduced by Bavula. We extend the results for the invariants of the Weyl algebra with respect to the symplectic action of a finite group, for the rings of invariant differential operators on quotient varieties, and invariants of certain generalized Weyl algebras under the linear actions. We show that the filter dimension of all above mentioned algebras equals 1., Comment: 15 pages

Two matrix vector spaces $V,W\subset \mathbb C^{n\times n}$ are said to be equivalent if $SVR=W$ for some nonsingular $S$ and $R$. These spaces are congruent if $R=S^T$. We prove that if all matrices in $V$ and $W$ are symmetric, or all matrices in $V$ and $W$ are skew-symmetric, then $V$ and $W$ are congruent if and only if they are equivalent. Let $F: U\times\dots\times U\to V$ and $G: U'\times\dots\times U'\to V'$ be symmetric or skew-symmetric $k$-linear maps over $\mathbb C$. If there exists a set of linear bijections $\varphi_1,\dots,\varphi_k:U\to U'$ and $\psi:V\to V'$ that transforms $F$ to $G$, then there exists such a set with $\varphi_1=\dots=\varphi_k$., Comment: 18 pages

We classify all simple strong Harish-Chandra modules for the Lie superalgebra $W(m,n)$. We show that every such module is either strongly cuspidal or a module of the highest weight type. We construct tensor modules for $W(m,n)$, which are parametrized by simple finite-dimensional $gl(m,n)$-modules and show that every simple strongly cuspidal $W(m,n)$-module is a quotient of a tensor module. Finally, we realize modules of the highest weight type as simple quotients of the generalized Verma modules induced from tensor modules for $W(m-1,n)$., Comment: After this paper was completed, we learned about a recent preprint by Yaohui Xu and Rencai Lu, arXiv:2001.04089 [math.RT], where the same results were obtained

The lower transcendence degree, introduced by J. J Zhang, is an important non-commutative invariant in ring theory and non-commutative geometry strongly connected to the classical Gelfand-Kirillov transcendence degree. For LD-stable algebras, the lower transcendence degree coincides with the Gelfand-Kirillov dimension. We show that the following algebras are LD-stable and compute their lower transcendence degrees: rings of differential operators of affine domains, universal enveloping algebras of finite dimensional Lie superalgebras, symplectic reflection algebras and their spherical subalgebras, finite $W$-algebras of type $A$, generalized Weyl algebras over Noetherian domain (under a mild condition), some quantum groups. We show that the lower transcendence degree behaves well with respect to the invariants by finite groups, and with respect to the Morita equivalence. Applications of these results are given., Comment: 19 pages

We construct new families of positive energy representations of affine vertex algebras together with their free field realizations by using localization technique. We introduce the twisting functor T_\alpha on the category of modules over affine Kac--Moody algebra \widehat{g}_\kappa of level \kappa for any positive root \alpha of g, and the Wakimoto functor from a certain category of g-modules to the category of smooth \widehat{g}_\kappa-modules. These two functors commute and the image of the Wakimoto functor consists of relaxed Wakimoto \widehat{g}_\kappa-modules. In particular, applying the twisting functor T_\alpha to the relaxed Wakimoto \widehat{g}_\kappa-module whose top degree component is isomorphic to the Verma g-module M^g_b(\lambda), we obtain the relaxed Wakimoto \widehat{g}_\kappa-module whose top degree component is isomorphic to the \alpha-Gelfand--Tsetlin g-module W^g_b(\lambda, \alpha). We show that the relaxed Verma module and relaxed Wakimoto module whose top degree components are such \alpha-Gelfand--Tsetlin modules, are isomorphic generically. This is an analogue of the result of E.Frenkel for Wakimoto modules both for critical and non-critical level. For a parabolic subalgebra p of g we construct a large family of admissible g-modules as images under the twisting functor of generalized Verma modules induced from p. In this way, we obtain new simple positive energy representations of simple affine vertex algebras., Comment: 45 pages, 2 figures

We explicitly construct, in terms of Gelfand--Tsetlin tableaux, a new family of simple positive energy representations for the simple affine vertex algebra V_k(sl_{n+1}) in the minimal nilpotent orbit of sl_{n+1}. These representations are quotients of induced modules over the affine Kac-Moody algebra of sl_n+1 and include in particular all admissible simple highest weight modules and all simple modules induced from sl_2. Any such simple module in the minimal nilpotent orbit has bounded weight multiplicities., Comment: 29 pages, to appear in IMRN

We address the problem of classifying of irreducible Gelfand-Tsetlin modules for gl(m|n) and show that it reduces to the classification of Gelfand-Tsetlin modules for the even part. We also give an explicit tableaux construction and the irreducibility criterion for the class of quasi typical and quasi covariant Gelfand-Tsetlin modules which includes all essentially typical and covariant tensor finite dimensional modules. In the quasi typical case new irreducible representations are infinite dimensional gl(m|n)-modules which are isomorphic to the parabolically induced (Kac) modules., Comment: to appear in Math. Research Letters

Let $V$ be a vector space over a field $\mathbb F$ with scalar product given by a nondegenerate sesquilinear form whose matrix is diagonal in some basis. If $\mathbb F=\mathbb C$, then we give canonical matrices of isometric and selfadjoint operators on $V$ using known classifications of isometric and selfadjoint operators on a complex vector space with nondegenerate Hermitian form. If $\mathbb F$ is a field of characteristic different from $2$, then we give canonical matrices of isometric, selfadjoint, and skewadjoint operators on $V$ up to classification of symmetric and Hermitian forms over finite extensions of $\mathbb F$., Comment: 21 pages

Realization by linear vector fields is constructed for any Lie algebra which admits a biorthogonal system and for its any suitable representation. The embedding into Lie algebras of linear vector fields is analogous to the classical Jordan-Schwinger map. A number of examples of such Lie algebras of linear vector fields is computed. In particular, we obtain examples of the twisted Heisenberg-Virasoro Lie algebra and the Schr\"odinger-Virasoro Lie algebras among others. More generally, we construct an embedding of an arbitrary locally convex topological algebra into the Cuntz algebra., Comment: 19 pages

Our purpose is to give new proofs of several known results about perturbations of matrix pencils. Andrzej Pokrzywa (1986) described the closure of orbit of a Kronecker canonical pencil $A-\lambda B$ in terms of inequalities with pencil invariants. In more detail, Pokrzywa described all Kronecker canonical pencils $K-\lambda L$ such that each neighborhood of $A-\lambda B$ contains a pencil whose Kronecker canonical form is $K-\lambda L$. Another solution of this problem was given by Klaus Bongartz (1996) by methods of representation theory. We give a direct and constructive proof of Pokrzywa's theorem. We reduce its proof to the cases of matrices under similarity and of matrix pencils $P-\lambda Q$ that are direct sums of two indecomposable Kronecker canonical pencils. We calculate the Kronecker forms of all pencils in a neighborhood of such a pencil $P-\lambda Q$. In fact, we calculate the Kronecker forms of only those pencils that belong to a miniversal deformation of $P-\lambda Q$, which is sufficient since all pencils in a neighborhood of $P-\lambda Q$ are reduced to them by smooth strict equivalence transformations.

We study non-standard Verma type modules over the Kac-Moody queer Lie superalgebra $\mathfrak{q}(n)^{(2)}$. We give a sufficient condition under which such modules are irreducible. We also give a classification of all irreducible diagonal $\mathbb{Z}$-graded modules over certain Heisenberg Lie superalgebras contained in $\mathfrak{q}(n)^{(2)}$., Comment: Minor corrections, published version

We associate to an arbitrary positive root $\alpha$ of a complex semisimple finite-dimensional Lie algebra $\mfrak{g}$ a twisting endofunctor $T_\alpha$ of the category of $\mfrak{g}$-modules. We apply this functor to generalized Verma modules in the category $\mcal{O}(\mfrak{g})$ and construct a family of $\alpha$-Gelfand--Tsetlin modules with finite $\Gamma_\alpha$-multiplicities, where $\Gamma_{\alpha}$ is a commutative $\C$-subalgebra of the universal enveloping algebra of $\mfrak{g}$ generated by a Cartan subalgebra of $\mfrak{g}$ and by the Casimir element of the $\mfrak{sl}(2)$-subalgebra corresponding to the root $\alpha$. This covers classical results of Andersen and Stroppel when $\alpha$ is a simple root and previous results of the authors in the case when $\mfrak{g}$ is a complex simple Lie algebra and $\alpha$ is the maximal root of $\mfrak{g}$. The significance of constructed modules is that they are Gelfand--Tsetlin modules with respect to any commutative $\C$-subalgebra of the universal enveloping algebra of $\mfrak{g}$ containing $\Gamma_\alpha$. Using the Beilinson--Bernstein correspondence we give a geometric realization of these modules together with their explicit description. We also identify a tensor subcategory of the category of $\alpha$-Gelfand--Tsetlin modules which contains constructed modules as well as the category $\mcal{O}(\mfrak{g})$.

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 simple Gelfand-Tsetlin sl(3)-modules as subquotionets of localized Gelfand-Tsetlin E_{21}-injective modules., Comment: Preliminary version. 97 pages, 46 pictures

We develop a general framework for studying relative weight representations for certain pairs consisting of an associative algebra and a commutative subalgebra. Using these tools we describe projective and simple weight modules for quantum Weyl algebras for generic values of deformation parameters. We consider two quantum versions: one by Maltsiniotis and the other one by Akhavizadegan and Jordan.

Recently we defined imaginary crystal bases for $U_q(\widehat{\mathfrak{sl}(2)})$- modules in category $\mathcal O^q_{\text{red,im}}$ and showed the existence of such bases for reduced quantized imaginary Verma modules for $U_q(\widehat{\mathfrak{sl}(2)})$. In this paper we show the existence of imaginary crystal basis for any object in the category $\mathcal O^q_{\text{red,im}}$., Comment: 13 pages. arXiv admin note: text overlap with arXiv:1509.00865, arXiv:0903.1062

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 simple socles of the universal tableaux modules. We also prove that every simple Verma module appears as a socle of a universal tableaux module and hence obtain a description of the essential supports of all simple Verma modules. As a consequence, we prove the Strong Futorny-Ovsienko Conjecture on the sharpness of the upper bounds of the Gelfand-Tsetlin multiplicities. In addition we give a very explicit description of the support and essential support of the simple singular Verma module $M(-\rho)$, Comment: 25 pages, some changes made to Introduction in the new version

In representation theory, a classification problem is called wild if it contains the problem of classifying matrix pairs up to simultaneous similarity. The latter problem is considered as hopeless; it contains the problem of classifying an arbitrary finite system of vector spaces and linear mappings between them. We prove that an analogous "universal" problem in the theory of tensors of order at most 3 over an arbitrary field is the problem of classifying three-dimensional arrays up to equivalence transformations \[ [a_{ijk}]_{i=1}^{m}\,{}_{j=1}^{n}\,{}_{k=1}^{t}\ \mapsto\ \Bigl[ \sum_{i,j,k} a_{ijk}u_{ii'} v_{jj'}w_{kk'}\Bigr]{}_{i'=1}^{m}\,{}_{j'=1}^{n}\,{}_{k'=1}^{t} \] in which $[u_{ii'}]$, $[v_{jj'}]$, $[w_{kk'}]$ are nonsingular matrices: this problem contains the problem of classifying an arbitrary system of tensors of order at most three., Comment: 38 pages

We construct explicitly a large family of Gelfand-Tsetlin modules for an arbitrary finite W-algebra of type A and establish their irreducibility. A basis of these modules is formed by the Gelfand-Tsetlin tableaux whose entries satisfy certain admissible sets of relations. Characterization and an effective method of constructing such admissible relations are given. In the case of the Yangian of gl_n we prove the sufficient condition for the irreducibility of the tensor product of two highest weight relation modules and establish irreducibility of any number of highest weight relation modules with generic highest weights. This extends the results of Molev to infinite dimensional highest modules.

We address the Noncommutative Noether's Problem on the invariants of Weyl fields for linear actions of finite groups. We prove that if the variety An(k)/G is rational then the Noncommutative Noether's Problem is positively solved for G and any field k of characteristic zero. In particular, this gives positive solution for all pseudo-reflections groups, for the alternating groups (n = 3,4,5) and for any finite group when n = 3. Alternative proofs are given for the complex field and for all pseudo-reflections groups. In the later case an effective algorithm of finding the Weyl generators is described. We also study birational equivalence for the rings of invariant differential operators on complex affine irreducible varieties., Comment: Fourth version improves Theorem 1.2. Appendix removed as an error was found in the proof of Thm. 8.3; Lemma 8.1 however is correct

We prove that an invariant subalgebra A_n^W of the Weyl algebra A_n is a Galois order over an adequate commutative subalgebra \Gamma when W is a two-parameters irreducible unitary reflection group G(m,1,n), m\geq 1, n\geq 1, including the Weyl group of type B_n, or alternating group, or the product of n copies of a cyclic group of fixed finite order. Earlier this was established for the symmetric group by the authors. In each of the cases above, except for the alternating groups, we show that A_n^W is free as a right (left) \Gamma-module. Similar results are established for the algebra of W-invariant differential operators on the n-dimensional torus where W is a symmetric group S_n or orthogonal group of type B_n or D_n. As an application of our technique we prove the quantum Gelfand-Kirillov conjecture for U_q(sl_2), the first Witten deformation and the Woronowicz deformation.

We describe Borel and parabolic subalgebras of affine Lie superalgebras and study the Verma type modules associated to such subalgebras. We give necessary and sufficient conditions under which these modules are simple., Comment: Comments welcome

We construct realizations of quantum generalized Verma modules for U_q(sl_n(C)) by quan- tum differential operators. Taking the classical limit q ! 1 provides a realization of classical generalized Verma modules for sl_n(C) by differential operators.

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 $A$.

We introduce the notion of virtual endomorphisms of Lie algebras and use it as an approach for constructing self-similarity of Lie algebras. This is done in particular for a class of metabelian Lie algebras having homological type F Pn, which are variants of lamp-lighter groups. We establish several criteria when the existence of virtual endomorphism implies a self-similar Lie structure. Furthermore, we prove that the classical Lie algebra sln(k), where char(k) does not divide n affords non-trivial faithful self-similarity.

In the present paper we describe a new class of Gelfand--Tsetlin modules for an arbitrary complex simple finite-dimensional Lie algebra g and give their geometric realization as the space of delta-functions" on the flag manifold G/B supported at the 1-dimensional submanifold. When g=sl(n) (or gl(n)) these modules form a subclass of Gelfand-Tsetlin modules with infinite dimensional weight subspaces. We discuss their properties and describe the simplicity criterion for these modules in the case of the Lie algebra sl(3,C).

We construct two examples of q-deformed classical Howe dual pairs (sl(2,C), sl(2,C)) and (sl(2,C), sl(n,C)). Moreover, we obtain a noncommutative version of the first fundamental theorem of classical invariant theory. Our approach to these duality differs from the paper of Lehrer-Zhang-Zhang. Furthermore, we solve the tensor product decomposition problem for Verma modules over U_q(sl(2,C)) provided q is not a root of unity.