Parabolic Verma Modules Research Articles

Walled Brauer algebras as idempotent truncations of level 2 cyclotomic quotients

We realize (via an explicit isomorphism) the walled Brauer algebra Br,t(δ) for arbitrary integral parameter δ as an idempotent truncation of a level two cyclotomic degenerate affine walled Brauer algebra. The latter arises naturally in Lie theory as the endomorphism ring of so-called mixed tensor products, i.e. of a parabolic Verma module tensored with some copies of the natural representation and its dual. The result provides a method to construct central elements in the walled Brauer algebra, and also implies the Koszulity of the walled Brauer algebras if δ≠0.

Modules for double affine Lie algebras

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.

Relaxed singular vectors, Jack symmetric functions and fractional level [formula omitted] models

The fractional level models are (logarithmic) conformal field theories associated with affine Kac–Moody (super)algebras at certain levels k∈Q. They are particularly noteworthy because of several longstanding difficulties that have only recently been resolved. Here, Wakimoto's free field realisation is combined with the theory of Jack symmetric functions to analyse the fractional level slˆ(2) models. The first main results are explicit formulae for the singular vectors of minimal grade in relaxed Wakimoto modules. These are closely related to the minimal grade singular vectors in relaxed (parabolic) Verma modules. Further results include an explicit presentation of Zhu's algebra and an elegant new proof of the classification of simple relaxed highest weight modules over the corresponding vertex operator algebra. These results suggest that generalisations to higher rank fractional level models are now within reach.

Bosonic Ghosts at c = 2 as a Logarithmic CFT

Motivated by Wakimoto free field realisations, the bosonic ghost system of central charge $c=2$ is studied using a recently proposed formalism for logarithmic conformal field theories. This formalism addresses the modular properties of the theory with the aim being to determine the (Grothendieck) fusion coefficients from a variant of the Verlinde formula. The key insight, in the case of bosonic ghosts, is to introduce a family of parabolic Verma modules which dominate the spectrum of the theory. The results include S-transformation formulae for characters, non-negative integer Verlinde coefficients, and a family of modular invariant partition functions. The logarithmic nature of the corresponding ghost theories is explicitly verified using the Nahm-Gaberdiel-Kausch fusion algorithm.

On Dynamical Adjoint Functor

We give an explicit formula relating the dynamical adjoint functor and dynamical twist over nonalbelian base to the invariant pairing on parabolic Verma modules. As an illustration, we give explicit $U(sl(n))$- and $U_\hbar(sl(n))$-invariant star product on projective spaces.

Recursive properties of branching and BGG resolution

Recurrent relations for branching coefficients are based on a special type of singular element decomposition. We show that this decomposition can be used to construct the parabolic Verma modules and finally to obtain the generalized Weyl-Verma formulas for characters. We demonstrate how branching coefficients can determine the generalized BGG resolution sequence.

On the Ext algebras of parabolic Verma modules and [formula omitted]-structures

We study the Ext-algebra of the direct sum of all parabolic Verma modules in the principal block of the Bernstein–Gelfand–Gelfand category O for the Hermitian symmetric pair ( g l n + m , g l n ⊕ g l m ) and present the corresponding quiver with relations for the cases n = 1 , 2 . The Kazhdan–Lusztig combinatorics is used to deduce a general vanishing result for the higher multiplications in the A ∞ -structure of a minimal model. An example of higher multiplications with non-vanishing m 3 is included.