Parabolic Verma Modules Research Articles

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.

Crystal base of the negative half of the quantum superalgebra [formula omitted

We construct a crystal base of Uq(gl(m|n))−, the negative half of the quantum superalgebra Uq(gl(m|n)). We give a combinatorial description of the associated crystal Bm|n(∞), which is equal to the limit of the crystals of the (q-deformed) Kac modules K(λ). We also construct a crystal base of a parabolic Verma module X(λ) associated with the subalgebra Uq(gl(0|n)), and show that it is compatible with the crystal base of Uq(gl(m|n))− and the Kac module K(λ) under the canonical embedding and projection of X(λ) to Uq(gl(m|n))− and K(λ), respectively.

Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions

We obtain BGG-type formulas for transfer matrices of irreducible finite-dimensional representations of the classical Lie algebras $\mathfrak{g}$, whose highest weight is a multiple of a fundamental one and which can be lifted to the representations over the Yangian $Y(\mathfrak{g})$. These transfer matrices are expressed in terms of transfer matrices of certain infinite-dimensional highest weight representations (such as parabolic Verma modules and their generalizations) in the auxiliary space. We further factorise the corresponding infinite-dimensional transfer matrices into the products of two Baxter $Q$-operators, arising from our previous study (arXiv:2001.04929, arXiv:2104.14518) of the degenerate Lax matrices. Our approach is crucially based on the new BGG-type resolutions of the finite-dimensional $\mathfrak{g}$-modules, which naturally arise geometrically as the restricted duals of the Cousin complexes of relative local cohomology groups of ample line bundles on the partial flag variety $G/P$ stratified by $B_{-}$-orbits.

Jantzen coefficients and simplicity of parabolic Verma modules

We systematically study the Jantzen coefficients. They share some properties with the leading coefficients of Kazhdan-Lusztig polynomials (the μ-functions) and are relatively easy to calculate. We develop a reduction process which reduces the computation of the Jantzen coefficients to the case of basic parabolic Verma modules. Through a classification of such modules, all the Jantzen coefficients for classical Lie algebras are obtained up to a sign. As an application of our results, we give refinements of Jantzen's simplicity criteria.

A note on parabolic Verma module homomorphisms over Kac-Moody algebras

We consider a general parabolic category O S over symmetrizable Kac-Moody algebras. We introduce a reduction rule of hom-spaces between parabolic Verma modules over different Kac-Moody algebras which yield some applications on parabolic Verma module homomorphisms .

2-Verma modules

Abstract 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 modules.

Schur–Weyl duality, Verma modules, and row quotients of Ariki–Koike algebras

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 particular case of our algebras.

2-Verma modules and the Khovanov–Rozansky link homologies

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.

A criterion for irreducibility of parabolic baby Verma modules of reductive Lie algebras

Let G be a connected, reductive algebraic group over an algebraically closed field k of prime characteristic p and g=Lie(G). In this paper, we study representations of g with a p-character χ of standard Levi form. When g is of type An,Bn,Cn or Dn, a sufficient condition for the irreducibility of standard parabolic baby Verma g-modules is obtained. This partially answers a question raised by Friedlander and Parshall (1990) in [2]. Moreover, as an application, in the special case that g is of type An or Bn, and χ lies in the sub-regular nilpotent orbit, we recover a result of Jantzen (1999) in [6].

Weyl-Kac character formula for affine Lie algebras in Deligne's categories

We study the characters of simple modules in the parabolic BGG category of the affine Lie algebra in Deligne's category. More specifically, we take the limit of the Weyl-Kac formula to compute the character of the irreducible quotient L(X,k) of the parabolic Verma module M(X,k) of level k, where X is an indecomposable object of Deligne's category Re_p(GLt), Re_p(Ot), or Re_p(Spt), under conditions that the highest weight of X plus the level gives a fundamental weight, t is transcendental, and the base field k has characteristic 0. We compare our result to the partial result in [7, Problem 6.2], and evaluate the characters to the categorical dimensions to get a categorical interpretation of the Nekrasov-Okounkov hook length formula, [12, Formula (6.12)].

Solutions of super Knizhnik–Zamolodchikov equations

We establish an explicit bijection between the sets of singular solutions of the (super) KZ equations associated with the Lie superalgebra, of infinite rank, of type \(\mathfrak {a}, \mathfrak {b},\mathfrak {c},\mathfrak {d}\) and with the corresponding Lie algebra. As a consequence, the singular solutions of the super KZ equations associated with the classical Lie superalgebra, of finite rank, of type \(\mathfrak {a}, \mathfrak {b},\mathfrak {c},\mathfrak {d}\) for the tensor product of certain parabolic Verma modules (resp., irreducible modules) are obtained from the singular solutions of the KZ equations for the tensor product of the corresponding parabolic Verma modules (resp., irreducible modules) over the corresponding Lie algebra of sufficiently large rank, and vice versa. The analogous results for some special kinds of trigonometric (super) KZ equations are obtained.

Polology of Superconformal Blocks

We systematically classify all possible poles of superconformal blocks as a function of the scaling dimension of intermediate operators, for all superconformal algebras in dimensions three and higher. This is done by working out the recently-proven irreducibility criterion for parabolic Verma modules for classical basic Lie superalgebras. The result applies to correlators for external operators of arbitrary spin, and indicates presence of infinitely many short multiplets of superconformal algebras, most of which are non-unitary. We find a set of poles whose positions are shifted by linear in $\mathcal{N}$ for $\mathcal{N}$-extended supersymmetry. We find an interesting subtlety for 3d $\mathcal{N}$-extended superconformal algebra with $\mathcal{N}$ odd associated with odd non-isotropic roots. We also comment on further applications to superconformal blocks.

Equivariant Vector Bundles Over Quantum Projective Spaces

We construct equivariant vector bundles over quantum projective spaces making use of parabolic Verma modules over the quantum general linear group. Using an alternative realization of the quantized coordinate ring of projective space as a subalgebra in the algebra of functions on the quantum group, we reformulate quantum vector bundles in terms of quantum symmetric pairs. In this way, we prove complete reducibility of modules over the corresponding coideal stabilizer subalgebras, via the quantum Frobenius reciprocity.

Tensor ideals, Deligne categories and invariant theory

We derive some tools for classifying tensor ideals in monoidal categories. We use these results to classify tensor ideals in Deligne’s universal categories $${\underline{\mathrm{Rep}}} O_\delta ,{\underline{\mathrm{Rep}}} GL_\delta $$ and $${\underline{\mathrm{Rep}}} P$$ . These results are then used to obtain new insight into the second fundamental theorem of invariant theory for the algebraic supergroups of types A, B, C, D, P. We also find new short proofs for the classification of tensor ideals in $${\underline{\mathrm{Rep}}} S_t$$ and in the category of tilting modules for $${\text {SL}}_2(\Bbbk )$$ with $$\mathrm{char}(\Bbbk )>0$$ and for $$U_q(\mathfrak {sl}_2)$$ with q a root of unity. In general, for a simple Lie algebra $$\mathfrak {g}$$ of type ADE, we show that the lattice of such tensor ideals for $$U_q(\mathfrak {g})$$ corresponds to the lattice of submodules in a parabolic Verma module for the corresponding affine Kac–Moody algebra.

Determinant formula for parabolic Verma modules of Lie superalgebras

We prove a determinant formula for a parabolic Verma module of a contragredient finite-dimensional Lie superalgebra, previously conjectured by the second author. Our determinant formula generalizes the previous results of Jantzen for a parabolic Verma module of a (non-super) Lie algebra, and of Kac concerning a (non-parabolic) Verma module for a Lie superalgebra. The resulting formula is expected to have a variety of applications in the study of higher-dimensional supersymmetric conformal field theories. We also discuss irreducibility criteria for the Verma module.

Comments on determinant formulas for general CFTs

We point out that the determinant formula for a parabolic Verma module plays a key role in the study of (super)conformal field theories and in particular their (super)conformal blocks. The determinant formula is known from the old work of Jantzen for bosonic conformal algebras, and we present a conjecture for superconformal algebras. The application of the formula includes derivation of the unitary bound and recursion relations for conformal blocks.

Recursion relations for conformal blocks

In the context of conformal field theories in general space-time dimension, we find all the possible singularities of the conformal blocks as functions of the scaling dimension $\Delta$ of the exchanged operator. In particular, we argue, using representation theory of parabolic Verma modules, that in odd spacetime dimension the singularities are only simple poles. We discuss how to use this information to write recursion relations that determine the conformal blocks. We first recover the recursion relation introduced in 1307.6856 for conformal blocks of external scalar operators. We then generalize this recursion relation for the conformal blocks associated to the four point function of three scalar and one vector operator. Finally we specialize to the case in which the vector operator is a conserved current.

Generalised Jantzen filtration of exceptional Lie superalgebras

Let g be an exceptional Lie superalgebra, and let p be the maximal parabolic subalgebra which contains the distinguished Borel subalgebra and has a purely even Levi subalgebra. For any parabolic Verma module in the parabolic category Op, it is shown that the Jantzen filtration is the unique Loewy filtration, and the decomposition numbers of the layers of the filtration are determined by the coefficients of inverse Kazhdan–Lusztig polynomials. An explicit description of the submodule lattices of the parabolic Verma modules is given, and formulae for characters and dimensions of the finite-dimensional simple modules are obtained.

Faces and maximizer subsets of highest weight modules

In this paper we study general highest weight modules Vλ over a complex finite-dimensional semisimple Lie algebra g. We present three formulas for the set of weights of a large family of modules Vλ, which include but are not restricted to all simple modules and all parabolic Verma modules. These formulas are direct and do not involve cancellations, and were not previously known in the literature. Our results extend the notion of the Weyl polytope to general highest weight g-modules Vλ.We also show that for all simple modules, the convex hull of the weights is a WJ-invariant polyhedron for some parabolic subgroup WJ. We compute its vertices, faces, and symmetries – more generally, we also do this for all parabolic Verma modules, and for all modules Vλ with highest weight λ not on a simple root hyperplane. To show our results, we extend the notion of convexity to arbitrary additive subgroups A⊂(R,+) of coefficients. Our techniques enable us to completely classify “weak A-faces” of the support sets wt(Vλ), in the process extending classical results of Satake, Borel–Tits, Vinberg, and Casselman, as well as modern variants by Chari–Dolbin–Ridenour and Cellini–Marietti, to general highest weight modules.