Verma Modules Research Articles

Graded extensions of Verma modules

Abstract In this paper, we investigate extensions between graded Verma modules in the Bernstein–Gelfand–Gelfand category $\mathcal{O}$ . In particular, we determine exactly which information about extensions between graded Verma modules is given by the coefficients of the R-polynomials. We also give some upper bounds for the dimensions of graded extensions between Verma modules in terms of Kazhdan–Lusztig combinatorics. We completely determine all extensions between Verma module in the regular block of category $\mathcal{O}$ for $\mathfrak{sl}_4$ and construct various “unexpected” higher extensions between Verma modules.

Remarks on q-difference opers arising from quantum toroidal algebras

Abstract We propose a conjectural correspondence between 
the spectra of the Bethe algebra for the quantum toroidal $\mathfrak{gl}_2$
algebra on relaxed Verma modules, 
and $q$-hypergeometric opers with apparent singularities.
We introduce alongside 
the notion of apparent singularities for linear $q$-difference 
operators and discuss some of their properties.
We also touch on a generalization to $\mathfrak{gl}_n$.

A note on the Loewy lengths of baby Verma modules for modular Lie algebras

Let G be a connected reductive algebraic group over an algebraically closed field k of prime characteristic p, and g=Lie(G). We study the representations of the reductive Lie algebra g with p-character χ of standard Levi-form in this note. We obtain the similar results about the translation functor and the wall-crossing functor of simple modules parallelling to the representations of algebraic groups (cf. [16, II.Lem.7.20]). Moreover, we get the Loewy lengths of baby Verma modules provided the Vogan Conjecture holds.

Representations of the rational Cherednik algebra [formula omitted] in positive characteristic

We study the rational Cherednik algebra Ht,c(S3,h) of type A2 in positive characteristic p, and its irreducible category O representations Lt,c(τ). For every possible value of p,t,c, and τ we calculate the Hilbert polynomial and the character of Lt,c(τ), and give explicit generators of the maximal proper graded submodule of the Verma module.

Quantum Sugawara operators in type A

The quantum Sugawara operators associated with a simple Lie algebra g are elements of the center of a completion of the quantum affine algebra Uq(gˆ) at the critical level. By the foundational work of Reshetikhin and Semenov-Tian-Shansky (1990), such operators occur as coefficients of a formal Laurent series ℓV(z) associated with every finite-dimensional representation V of the quantum affine algebra. As demonstrated by Ding and Etingof (1994), the quantum Sugawara operators generate all singular vectors in generic Verma modules over Uq(gˆ) at the critical level and give rise to a commuting family of transfer matrices. Furthermore, as observed by E. Frenkel and Reshetikhin (1999), the operators are closely related with the q-characters and q-deformed W-algebras via the Harish-Chandra homomorphism.We produce explicit quantum Sugawara operators for the quantum affine algebra of type A which are associated with primitive idempotents of the Hecke algebra and parameterized by Young diagrams. This opens a way to understand all the related objects via their explicit constructions. We consider one application by calculating the Harish-Chandra images of the quantum Sugawara operators. The operators act by scalar multiplication in the q-deformed Wakimoto modules and we calculate the eigenvalues by identifying them with the Harish-Chandra images.

Shapovalov elements for U q ( s l ( N + 1 ) )

For a simple Lie algebra, Shapovalov elements give rise to highest weight vectors in Verma modules. The usual construction of these elements uses induction on the length of a certain Weyl group element. If g = s l ( N + 1 ) explicit expressions for Shapovalov elements were given in I. M. Musson [Explicit expressions for Shapovalov elements in type A, J. Algebra 623 (2023), 358–394]. Here we adapt the argument to the quantized enveloping algebra of g .

Join operation for the Bruhat order and Verma modules

We observe that the join operation for the Bruhat order on a Weyl group agrees with the intersections of Verma modules in type A. The statement is not true in other types, and we propose a weaker correspondence. Namely, we introduce distinguished subsets of the Weyl group on which the join operation conjecturally agrees with the intersections of Verma modules. We also relate our conjecture with a statement about the socles of the cokernels of inclusions between Verma modules. The latter determines the first Ext space between a simple module and a Verma module. We give a conjectural complete description of such socles which we verify in a number of cases. Along the way, we determine the poset structure of the join-irreducible elements in Weyl groups and obtain closed formulae for certain families of Kazhdan–Lusztig polynomials.

Representations of the [formula omitted]-series related to the q-analog Virasoro-like Lie algebra

In this paper, we study the representation of an infinite-dimensional Lie algebra C related to the q-analog Virasoro-like Lie algebra. We give the necessary and sufficient conditions for the highest weight irreducible module V(ϕ) of C to be a Harish-Chandra module. We prove that the Verma C-module V¯(ϕ) is either irreducible or has the corresponding irreducible highest weight C-module V(ϕ) that is a Harish-Chandra module. We also give the maximal proper submodule of the Verma module V¯(ϕ) and the e-character of the irreducible highest weight C-module V(ϕ) when the highest weight ϕ satisfies some natural conditions. Furthermore, we give the classification of the Harish-Chandra C-modules with nontrivial central charge.

Branching problem for tensoring two Verma modules and its application to differential symmetry breaking operators

Kobayashi–Pevzner discovered in [Differential symmetry breaking operators: II. Rankin–Cohen operators for symmetric pairs, Selecta Math. (N.S.) 22(2) (2016) 847–911, MR 3477337] that the failure of the multiplicity-one property in the fusion rule of Verma modules of [Formula: see text] occurs exactly when the Rankin–Cohen brackets vanish, and classified all the corresponding parameters. In this paper, we provide yet another characterization for these parameters, and give a precise description of indecomposable components of the tensor product. Furthermore, we discuss when the tensor products of two Verma modules are isomorphic to each other for semisimple Lie algebras [Formula: see text].

Smooth modules over the N = 1 Bondi–Metzner–Sachs superalgebra

In this paper, we present a determinant formula for a contravariant form on Verma modules over the [Formula: see text] Bondi–Metzner–Sachs (BMS) superalgebra. This formula establishes a necessary and sufficient condition for the irreducibility of the Verma modules. We then introduce and characterize a class of simple smooth modules that generalize both Verma and Whittaker modules over the [Formula: see text] BMS superalgebra. We also utilize the Heisenberg–Clifford vertex superalgebra to construct a free field realization for the [Formula: see text] BMS superalgebra. This free field realization allows us to obtain a family of natural smooth modules over the [Formula: see text] BMS superalgebra, which includes Fock modules and certain Whittaker modules.

Unified invariant of knots from homological braid action on Verma modules

AbstractWe re‐build the quantum unified invariant of knots 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, that is, semisimple and non‐semisimple invariants of knots constructed from quantum . We prove this last fact in our context that re‐proves (a generalization of) the famous Melvin–Morton–Rozansky conjecture first proved by Bar‐Natan and Garoufalidis. We find a symmetry of nicely generalizing the well‐known one of the Alexander polynomial, ADO polynomials also inherit this symmetry. It implies that quantum non‐semisimple invariants are not detecting knots' orientation. Using the homological definition of Verma modules we express as a generating sum of intersection pairing between fixed Lagrangians of configuration spaces of disks. Finally, we give a formula for using a generalized notion of determinant, that provides one for the ADO family. It generalizes that for the Alexander invariant.

On the intertwining differential operators from a line bundle to a vector bundle over the real projective space

We classify and construct SL(n,R)-intertwining differential operators D from a line bundle to a vector bundle over the real projective space RPn−1 by the F-method. This generalizes a classical result of Bol for SL(2,R). Further, we classify the K-type formulas for the kernel Ker(D) and image Im(D) of D. The standardness of the homomorphisms φ corresponding to the differential operators D between generalized Verma modules is also discussed.

Representations of the affine ageing algebra agê(1)

In this paper, we investigate the affine ageing algebra agê(1), which is a central extension of the loop algebra of the one-spatial ageing algebra age(1). Certain Verma-type modules including Verma modules and imaginary Verma modules of agê(1) are studied. Particularly, the simplicity of these modules are characterized and their irreducible quotient modules are determined. We also study the restricted modules of agê(1) which are also the modules of the affine vertex algebra arising from the one-spatial ageing algebra age(1). We present certain constructions of simple restricted agê(1)-modules and an explicit such example of simple restricted module via the Whittaker module of agê(1) is given.

Gelfand–Kirillov Dimension and Reducibility of Scalar Generalized Verma Modules for Classical Lie Algebras

Gelfand–Kirillov Dimension and Reducibility of Scalar Generalized Verma Modules for Classical Lie Algebras

Shapovalov elements of classical and quantum groups

Shapovalov elements θβ,m of the classical or quantized universal enveloping algebra of a simple Lie algebra g are parameterized by a positive root β and a positive integer m. They relate the highest vector of a reducible Verma module with highest vectors of its submodules. We obtain a factorization of θβ,m to a product of θβ,1 and calculate θβ,1 as a residue of a matrix element of the inverse Shapovalov form via a generalized Nagel-Moshinsky algorithm. This way we explicitly express θβ,m of a classical simple Lie algebra through the Cartan-Weyl basis in g. In the case of quantum groups, we give an analogous formulation through the entries of the R-matrix (quantum L-operator) in fundamental representations.

Unique Factorization for Tensor Products of Parabolic Verma Modules

Unique Factorization for Tensor Products of Parabolic Verma Modules

Crystal bases of parabolic Verma modules over the quantum orthosymplectic superalgebras

We show that there exists a unique crystal base of a parabolic Verma module over a quantum orthosymplectic superalgebra, which is induced from a q-analogue of a polynomial representation of a general linear Lie superalgebra.

Character formulas in category 𝒪_{𝓅}

Let O p \mathcal {O}_p denote the characteristic p > 0 p>0 version of the ordinary category O \mathcal {O} for a semisimple complex Lie algebra. In this paper we give some (formal) character formulas in O p \mathcal {O}_p . First we concentrate on the irreducible characters. Here we give explicit formulas for how to obtain all irreducible characters from the characters of the finitely many restricted simple modules as well as the characters of a small number of infinite dimensional simple modules in O p \mathcal {O}_p with specified highest weights. We next prove a strong linkage principle for Verma modules which allow us to split O p \mathcal {O}_p into a finite direct sum of linkage classes. There are corresponding translation functors and we use these to further cut down the set of irreducible characters needed for determining all others. Then we show that the twisting functors on O \mathcal {O} carry over to twisting functors on O p \mathcal {O}_p , and as an application we prove a character sum formula for Jantzen-type filtrations of Verma modules with antidominant highest weights. Finally, we record formulas relating the characters of the two kinds of tilting modules in O p \mathcal {O}_p .

Whittaker categories of quasi-reductive lie superalgebras and quantum symmetric pairs

Abstract We show that, for an arbitrary finite-dimensional quasi-reductive Lie superalgebra over ${\mathbb {C}}$ with a triangular decomposition and a character $\zeta $ of the nilpotent radical, the associated Backelin functor $\Gamma _\zeta $ sends Verma modules to standard Whittaker modules provided the latter exist. As a consequence, this gives a complete solution to the problem of determining the composition factors of the standard Whittaker modules in terms of composition factors of Verma modules in the category ${\mathcal {O}}$ . In the case of the ortho-symplectic Lie superalgebras, we show that the Backelin functor $\Gamma _\zeta $ and its target category, respectively, categorify a q-symmetrizing map and the corresponding q-symmetrized Fock space associated with a quasi-split quantum symmetric pair of type $AIII$ .

Homology of the complexes of finite Verma modules over CK6

We compute the homology of the first and third quadrants of the complexes of finite Verma modules over the annihilation superalgebra A(CK6)≅E(1,6), associated with the conformal superalgebra CK6, obtained in [6]. This computation allows us to explicitly realize the irreducible quotients of degenerate finite Verma modules over A(CK6).