We compute the algebras of self-extensions of the vacuum module and the Verma modules over an affine Kac-Moody algebra g^ in suitable categories of Harish-Chandra modules. We show that at the critical level these algebras are isomorphic to the algebras of differential forms on various spaces of opers associated to the Langlands dual Lie algebra of g, whereas away from the critical level they become trivial. These results rely on and generalize the description of the corresponding algebras of endomorphisms obtained by Feigin and Frenkel and the description of the corresponding graded versions due to Fishel, Grojnowski and Teleman.
In the present paper we investigate a new class of infinite-dimensional modules over the hyperalgebra of a semi-simple algebraic group in positive chararacteristic called quasi-Verma modules. We provide a purely algebraic construction of the global Grothendieck-Cousin complex corresponding to the standard line bumdle ${\mathcal L}(λ)$ on the Flag variety of the algebraic group stratified by Schubert cells. We prove that the complex consists of direct sums of quasi-Verma modules for the highest weights of the form $w\cdotλ$ for various elements $w$ of the Weyl group.
Using principal series Harish-Chandra modules, local cohomology with support in Schubert cells and twisting functors we construct certain modules parametrized by the Weyl group and a highest weight in the subcategory O of the category of representations of a complex semisimple Lie algebra. These are in a sense modules between a Verma module and its dual. We prove that the three different approaches lead to the same modules. Moreover, we demonstrate that they possess natural Jantzen type filtrations with corresponding sum formulae.
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.
Irreducibilities of Verma modules over a class of Block type Lie algebras are completely determined. The approach developed in the present paper can be used to deal with non-weight modules.
The character formula of any finite dimensional irreducible module for Lie superalgebra $\mathfrak{osp}(3|2m)$ is obtained in terms of characters of generalized Verma modules.
We classify and explicitly construct the embedding diagrams of Verma modules over the N=2 supersymmetric extension of the Virasoro algebra. The essential ingredient of the solution consists in drawing the distinction between two different types of submodules appearing in N=2 Verma modules. The problem is simplified by associating to every N=2 Verma module a relaxed Verma module over the affine algebra ^sl(2) with an isomorphic embedding diagram. We then make use of the mechanism according to which the structure of the N=2/relaxed-sl(2) embedding diagrams can be found knowing the standard embed…
The purpose of this paper is to find maximal and primitive elements of baby Verma modules for a quantum group of type $B_2$. As a consequence the composition factors of the baby Verma modules are determined. Similar approach can be used to find find maximal and primitive elements of Weyl modules for type $B_2$.
In our previous paper, we constructed an explicit GL(n)-equivariant quantization of the Kirillov--Kostant-Souriau bracket on a semisimple coadjoint orbit. In the present paper, we realize that quantization as a subalgebra of endomorphisms of a generalized Verma module. As a corollary, we obtain an explicit description of the annihilators of generalized Verma modules over U(gl(n)). As an application, we construct real forms of the quantum orbits and classify finite dimensional representations. We compute the non-commutative Connes index for basic homogenous vector bundles over the quantum orbit…
We construct a categorification of parabolic Verma modules for symmetrizable Kac-Moody algebras using KLR-like diagrammatic algebras. We show that our construction arises naturally from a dg-enhancement of the cyclotomic quotients of the KLR-algebras. As a consequence, we are able to recover the usual categorification of integrable modules. We also introduce a notion of dg-2-representation for quantum Kac--Moody algebras, and in particular of parabolic 2-Verma module.
Let g be an untwisted affine Kac-Moody algebra and M_J(lambda) a Verma-type module for g with J-highest integral weight lambda. We construct quantum Verma-type modules M_J^q(lambda) over the quantum group U_q(g), investigate their properties and show that M_J^q(lambda) is a true quantum deformation of M_J(ł) in the sense that the weight structure is preserved under the deformation. We also analyze the submodule structure of quantum Verma-type modules.
Let $\mathbb{F}$ be a field of characteristic 0, $G$ an additive subgroup of $\mathbb{F}$, $α\in \mathbb{F}$ satisfying $α\notin G, 2α\in G$. We define a class of infinite-dimensional Lie algebras which are called generalized Schrödinger-Virasoro algebras and use $\mathfrak{gsv}[G,α]$ to denote the one corresponding to $G$ and $α$. In this paper the automorphism group and irreducibility of Verma modules for $\mathfrak{gsv}[G,α]$ are completely determined.
Let $\mathfrak{g}$ be a reductive Lie algebra. We give a condition that ensures that the character of a generalized Verma module is well-behaved under a twisting functor. We show that a similar result holds for basic classical simple Lie superalgebras, and generalize a result from \cite{CM} about twisting Verma modules.
It is known that symmetric orbits in ${\bf g}^*$ for any simple Lie algebra ${\bf g}$ are equiped with a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the reduced Sklyanin bracket associated to the "canonical" R-matrix. We realize quantization of this Poisson pencil on $\cp$ type orbits (i.e. orbits in $sl(n+1)^*$ whose real compact form is $ CP^n$) by means of q-deformed Verma modules.
A class of generalized Verma modules over $\fr{sl}_{n+2}$ is constructed from $\fr{sl}_{n+1}$-modules which are $\uhn$-free modules of rank $1$. The necessary and sufficient conditions for these $\fr{sl}_{n+2}$-modules to be simple are determined. This leads to a class of new simple $\fr{sl}_{n+2}$-modules.
We consider imaginary Verma modules for quantum affine algebra $U_q(\hat{\mathfrak{sl}(2)})$ and construct Kashiwara type operators and the Kashiwara algebra $\mathcal K_q$. We show that a certain quotient $\mathcal N_q^-$ of $U_q(\hat{\mathfrak{sl}(2)})$ is a simple $\mathcal K_q$-module.
We construct a monomorphism of the De Rham complex of scalar multivalued meromorphic forms on the projective line, holomorphic on the complement to a finite set of points, to the chain complex of the Lie algebra of $sl_2$-valued algebraic functions on the same complement with coefficients in a tensor product of contragradient Verma modules over the affine Lie algebra $\hat{sl}_2$. We show that the existence of singular vectors in the Verma modules (the Malikov-Feigin-Fuchs singular vectors) is reflected in the new relations between the cohomology classes of logarithmic differential forms.
We construct vertex algebraic intertwining operators among certain generalized Verma modules for $\widehat{\mathfrak{sl}(2,\mathbb{C})}$ and calculate the corresponding fusion rules. Additionally, we show that under some conditions these intertwining operators descend to intertwining operators among one generalized Verma module and two (generally non-standard) irreducible modules. Our construction relies on the irreducibility of the maximal proper submodules of generalized Verma modules appearing in the Garland-Lepowsky resolutions of standard $\widehat{\mathfrak{sl}(2,\mathbb{C})}$-modules. W…
For any additive subgroup $G$ of an arbitrary field $F$ of characteristic zero, there corresponds a generalized Heisenberg-Virasoro algebra $L[G]$. Given a total order of $G$ compatible with its group structure, and any $h,h_I,c,c_I,c_{LI}\in F$, a Verma module $M(h,h_I,c,c_I,c_{LI})$ over $L[G]$ is defined. In the this note, the irreducibility of Verma modules $M(h,h_I,c,c_I,c_{LI})$ is completely determined.
We study the Verma modules M(mu(u)) over the Yangian Y(a) associated with a simple Lie algebra a. We give necessary and sufficient conditions for irreducibility of M(mu(u)). Moreover, regarding the simple quotient L(mu(u)) of M(mu(u)) as an a-module, we give necessary and sufficient conditions for finite-dimensionality of the weight subspaces of L(mu(u)).
We find sufficient conditions for the construction of vertex algebraic intertwining operators, among generalized Verma modules for an affine Lie algebra $\hat{\mathfrak{g}}$, from $\mathfrak{g}$-module homomorphisms. When $\mathfrak{g}=\mathfrak{sl}_2$, these results extend previous joint work with J. Yang, but the method used here is different. Here, we construct intertwining operators by solving Knizhnik-Zamolodchikov equations for three-point correlation functions associated to $\hat{\mathfrak{g}}$, and we identify obstructions to the construction arising from the possible non-existence of …
We consider imaginary Verma modules for quantum affine algebra $U_q(\hat{\mathfrak{g}})$, where $\hat{\mathfrak{g}}$ is of type 1 i.e. of non-twisted type, and construct Kashiwara type operators and the Kashiwara algebra $\mathcal K_q$. We show that a certain quotient $\mathcal N_q^-$ of $U_q(\hat{\mathfrak{g}})$ is a simple $\mathcal K_q$-module.
I present a simple dynamic programming algorithm for the evaluation of operators in a wide range of superconformal algebras. Special care is taken to describe the computation of the Gram matrix. A Mathematica package, Weaver.m, is provided that implements the algorithm.
Some natural hidden symmetries in the Verma modules over the Virasoro algebra are constructed in terms of geometric quantization. Their differential geometric meaning is established and their expression via $q_R$-conformal symmetries in the Verma modules over the Lie algebra $sl(2,C)$ is found. The analysis and the unraveling of the algebraic structure of these families of hidden symmetries are performed.
We consider two complexes. The first complex is the twisted de Rham complex of scalar meromorphic differential forms on projective line, holomorphic on the complement to a finite set of points. The second complex is the chain complex of the Lie algebra of $\mathfrak{sl}_2$-valued algebraic functions on the same complement, with coefficients in a tensor product of contragradient Verma modules over the affine Lie algebra $\hat{{\mathfrak{sl}_2}}$. In [Schechtman V., Varchenko A., Mosc. Math. J. 17 (2017), 787-802] a construction of a monomorphism of the first complex to the second was suggested …
We show that subsingular vectors exist in Verma modules over W(2,2), and present a subquotient structure of these modules. We prove conditions for irreducibility of a tensor product of intermediate series module with the highest weight module. Relations to intertwining operators over vertex operator algebra associated to W(2,2) is discussed. Also, we study a tensor product of intermediate series and highest weight module over the twisted Heisenberg-Virasoro algebra, and present series of irreducible modules with infinite-dimensional weight spaces.
We propose a universal IR formula for the protected three-sphere correlation functions of Higgs and Coulomb branch operators of ${\cal N}=4$ supersymmetric quantum field theories with massive, topologically trivial vacua.
The Drinfeld double D of the bosonization of a finite-dimensional Nichols algebra B(V) over a finite non-abelian group G is called a quantum group at a non-abelian group. We introduce Verma modules over such a quantum group D and prove that a Verma module has simple head and simple socle. This provides two bijective correspondences between the set of simple modules over D and the set of simple modules over the Drinfeld double D(G). As an example, we describe the lattice of submodules of the Verma modules over the quantum group at the symmetric group S3 attached to the 12-dimensional Fomin-Kiri…
Let B be the Lie algebra with basis {L_{i,j},C|i,j\in Z} and relations [L_{i,j},L_{k,l}]=((j+1)k-i(l+1))L_{i+k,j+l}+iδ_{i,-k}δ_{j+l,-2}C, [C,L_{i,j}]=0. It is proved that an irreducible highest weight B-module is quasifinite if and only if it is a proper quotient of a Verma module. For an additive subgroup G of the base field F, there corresponds to a Lie algebra B(G) of Block type. Given a totalorder \succ on G and a weight Λ, a Verma B(G)-module M(Λ,\succ) is defined. The irreducibility of M(Λ,\succ) is completely determined.
Given a weight of $sl(n,\mbb{C})$, we derive a system of variable-coefficient second-order linear partial differential equations that determines the singular vectors in the corresponding Verma module, and a differential-operator representation of the symmetric group $S_n$ on the related space of truncated power series. We prove that the solution space of the system of partial differential equations is exactly spanned by $\{\sgm(1)\mid \sgm\in S_n\}$. Moreover, the singular vectors of $sl(n,\mbb{C})$ in the Verma module are given by those $\sgm(1)$ that are polynomials. The well-known results o…
In [ZH], R.B. Zhang found a way to link certain formal deformations of the Lie algebra o(2l+1) and the Lie superalgebra osp(1,2l). The aim of this article is to reformulate the Zhang transformation in the context of the quantum enveloping algebras "à la" Drinfeld-Jimbo and to show how this construction can explain the main theorem of [GL2]: the annihilator of a Verma module over the Lie superalgebra osp(1,2l) is generated by its intersection with the centralizer of the even part of the enveloping algbra.
We explain how Queffelec-Sartori's construction of the HOMFLY-PT link polynomial can be interpreted in terms of parabolic Verma modules for $\mathfrak{gl}_{2n}$. Lifting the construction to the world of categorification, we use parabolic 2-Verma modules to give a higher representation theory construction of Khovanov-Rozansky's triply graded link homology.
The exceptional infinite-dimensional linearly compact simple Lie superalgebra ${E}(5,10)$, which Kac believes, is the algebra of symmetries of the ${SU}_{5}$ Grand Unified Model. In this paper, we give a proof of Kac and Rudakov's conjecture about the classification of all the degenerate generalized Verma module over ${E}(5,10)$. Also, we work out all the nontrivial singular vectors degree by degree. It is a potential that the representation theory of ${E}(5,10)$ will shed new light on various features of the the ${SU}_{5}$ Grand unified model.
Imaginary Verma modules, parabolic imaginary Verma modules, and Verma modules at level zero for double affine Lie algebras are constructed using three different triangular decompositions. Their relations are investigated, and several results are generalized from the affine Lie algebras. In particular, imaginary highest weight modules, integrable modules, and irreducibility criterion are also studied.
The Misra-Miwa $v$-deformed Fock space is a representation of the quantized affine algebra of type A. It has a standard basis indexed by partitions and the non-zero matrix entries of the action of the Chevalley generators with respect to this basis are powers of $v$. Partitions also index the polynomial Weyl modules for the quantum group $U_q(gl_N)$ as $N$ tends to infinity. We explain how the powers of $v$ which appear in the Misra-Miwa Fock space also appear naturally in the context of Weyl modules. The main tool we use is the Shapovalov determinant for a universal Verma module
The present paper contains two interrelated developments. First, are proposed new generalized Verma modules. They are called k-Verma modules, k\in N, and coincide with the usual Verma modules for k=1. As a vector space a k-Verma module is isomorphic to the symmetric tensor product of k copies of the universal enveloping algebra U(g^-), where g^- is the subalgebra of lowering generators in the standard triangular decomposition of a simple Lie algebra g = g^+ \oplus h \oplus g^- . The second development is the proposal of a procedure for the construction of multilinear intertwining differential …
We extend Lawrence's representations of the braid groups to relative homology modules, and we show that they are free modules over a Laurent polynomials ring. We define homological operators and we show that they actually provide a representation for an integral version for $U_q \mathfrak{sl}(2)$. We suggest an isomorphism between a given basis of homological modules and the standard basis of tensor products of Verma modules, and we show it to preserve the integral ring of coefficients, the action of $U_q \mathfrak{sl}(2)$, the braid group representations and their grading. This recovers an in…
We prove a Schur-Weyl duality between the quantum enveloping algebra of $\mathfrak{gl}_m$ and certain quotient algebras of Ariki-Koike algebras, which we give explicitly. The duality involves several algebraically independent parameters and is realized through the tensor product of a parabolic universal Verma module and a tensor power of the natural representation of $\mathfrak{gl}_m$. We also give a new presentation by generators and relations of the generalized blob algebras of Martin and Woodcock as well as an interpretation in terms of Schur-Weyl duality by showing they occur as a particul…
We re-build the quantum sl2 unified invariant of knots $F_{\infty}$ from braid groups' action on tensors of Verma modules. It is a two variables series having the particularity of interpolating both families of colored Jones polynomials and ADO polynomials, i.e. semi-simple and non semi-simple invariants of knots constructed from quantum sl2. We prove this last fact in our context which re-proves (a generalization of) the famous Melvin-Morton-Rozansky conjecture first proved by Bar-Natan and Garoufalidis. We find a symmetry of $F_{\infty}$ nicely generalizing the well known one of the Alexande…
We introduce a suitable adapted ordering for the twisted N=2 superconformal algebra (i.e. with mixed boundary conditions for the fermionic fields). We show that the ordering kernels for complete Verma modules have two elements and the ordering kernels for G-closed Verma modules just one. Therefore, spaces of singular vectors may be two-dimensional for complete Verma modules whilst for G-closed Verma modules they can only be one-dimensional. We give all singular vectors for the levels 1/2, 1, and 3/2 for both complete Verma modules and G-closed Verma modules. We also give explicit examples of d…
We study representations of nilpotent type nontrivial liftings of quantum linear spaces and their Drinfel'd quantum doubles. We construct a family of Verma- type modules in both cases and prove a parametrization theorem for simple modules. We compute the Loewy and socle series of Verma modules under a mild restriction on the datum of a lifting. We find bases and dimensions of simple modules.
We describe the structure of the irreducible highest weight modules for the twisted Heisenberg-Virasoro Lie algebra at level zero. We prove that such a module is either isomorphic to a Verma module or to a quotient of two Verma modules.
We prove the existence of a homomorphism of vertex algebras, from the vacuum Verma module over the loop algebra of an untwisted affine algebra, whose construction is analogous to that of the Feigin-Frenkel homomorphism from the vacuum Verma module at critical level over an affine algebra.
The Kac determinant for the Topological N=2 superconformal algebra is presented as well as a detailed analysis of the singular vectors detected by the roots of the determinants. In addition we identify the standard Verma modules containing `no-label' singular vectors (which are not detected directly by the roots of the determinants). We show that in standard Verma modules there are (at least) four different types of submodules, regarding size and shape. We also review the chiral determinant formula, for chiral Verma modules, adding new insights. Finally we transfer the results obtained to the …
We study homomorphisms between quantized generalized Verma modules $M(V_Λ)\stackrel{ϕ_{Λ,Λ_1}}{\rightarrow}M(V_{Λ_1})$ for ${\mathcal U}_q(su(n,n))$. There is a natural notion of degree for such maps, and if the map is of degree $k$, we write $ϕ^k_{Λ,Λ_1}$. We examine when one can have a series of such homomorphisms $ϕ^1_{Λ_{n-1},Λ_{n}} \circ ϕ^1_{Λ_{n-2}, Λ_{n-1}} \circ\cdots\circ ϕ^1_{Λ,Λ_1} = \textrm{Det}_q$, where $\textrm{Det}_q$ denotes the map $M(V_Λ)\ni p\rightarrow \textrm{Det}_q\cdot p\in M(V_{Λ_n})$. If, classically, $su(n,n)^{\mathbb C}={\mathfrak p}^-\oplus(su(n)\oplus su(n)\oplus…
In this paper we give some relationships among the Groebner-Shirshov bases in free associative algebras, free left modules and "double-free" left modules (free modules over a free algebra). We give the Chibrikov's Composition-Diamond lemma for modules and show that Kang-Lee's Composition-Diamond lemma follows from this lemma. As applications, we also deal with highest weight module over the Lie algebra $sl_2$, Verma module over a Kac-Moody algebra, Verma module over Lie algebra of coefficients of a free conformal algebra and the universal enveloping module for a Sabinin algebra.
By properly specializing the parameters irreducible modules of maximal dimension for the De Concini-Kac version of the Drinfeld-Jimbo quantum algebra in type $A$ may be transformed into modules over Lusztig's infinitesimal quantum algeba. Thus obtained modules have a simple socle and a simple head, and share the same dimension as the infinitesimal Verma modules. Despite these common features we find that they are never isomorphic to infinitesimal Verma modules unless they are irreducible. The same carry over to the modular setup for the special linear groups in positive characteristic.
Given a weight of sl(n), we derive a system of variable-coefficient second-order linear partial differential equations that determines the singular vectors in the corresponding Verma module. Moreover, we completely solve the system in a certain space of power series. The polynomial solutions correspond to the singular vectors in the Verma module. The well-known results of Verma, Bernstein-Gel'fand-Gel'fand and Jantzen for the case of $sl(n)$ are naturally included in our almost elementary approach of partial differential equations.
Verma modules of superconfomal algebras can have singular vector spaces with dimensions greater than 1. Following a method developed for the Virasoro algebra by Kent, we introduce the concept of adapted orderings on superconformal algebras. We prove several general results on the ordering kernels associated to the adapted orderings and show that the size of an ordering kernel implies an upper limit for the dimension of a singular vector space. We apply this method to the topological N=2 algebra and obtain the maximal dimensions of the singular vector spaces in the topological Verma modules: 0,…
We analyze several issues concerning the singular vectors of the Topological N=2 Superconformal algebra. First we investigate which types of singular vectors exist, regarding the relative U(1) charge and the BRST-invariance properties, finding four different types in chiral Verma modules and twenty-nine different types in complete Verma modules. Then we study the family structure of the singular vectors, every member of a family being mapped to any other member by a chain of simple transformations involving the spectral flows. The families of singular vectors in chiral Verma modules follow a u…
