Classifying relaxed highest-weight modules for admissible-level Bershadsky-Polyakov algebras

Abstract

A. The Bershadsky-Polyakov algebras are the minimal quantum hamiltonian reductions of the affine vertex algebras associated to sl 3 and their simple quotients have a long history of applications in conformal field theory and string theory. Their representation theories are therefore quite interesting. Here, we classify the simple relaxed highest-weight modules for all admissible but nonintegral levels, significantly generalising the known highest-weight classifications [1,2]. In particular, we prove that the simple Bershadsky-Polyakov algebras with admissible nonintegral k are always rational in category O, whilst they always admit nonsemisimple relaxed highest-weight modules unless k + 3 2 ∈ 0 .

Subregular W-algebras of type A

“…For = 1 and 2, the roots of ( , ) can be described using data from quantum hamiltonian reductions of certain highest-weight L k ( 2 )and L k ( 3 )-modules respectively. The = 2 case is essentially the second point in Theorem 4.20 from [16].…”

“…This connection is expressed beautifully in the associated variety of the subregular W-algebra [14]. In light of these and many more motivations, much recent work has been done to improve our understanding of the structure and representation theory of subregular W-algebras [15][16][17][18][19][20][21].In general, an understanding of W-algebras for nonregular nilpotents is highly desirable. One means of achieving this is to leverage the well-understood W-algebras to learn about others: It is strongly suspected that in addition to the usual quantum hamiltonian reduction, one can also perform a 'partial quantum hamiltonian reduction' between two W-algebras as long as their corresponding nilpotent orbits are related by a certain partial ordering [22].…”

“…This connection is expressed beautifully in the associated variety of the subregular W-algebra [14]. In light of these and many more motivations, much recent work has been done to improve our understanding of the structure and representation theory of subregular W-algebras [15][16][17][18][19][20][21].…”

Modularity of Bershadsky-Polyakov minimal models

“…For admissible levels with v > 2, these being the nondegenerate admissible levels, BP k is nonrational in the category of weight modules. This was shown [1] by combining the untwisted and twisted Zhu algebras of BP k with Arakawa's results [8] on minimal quantum hamiltonian reductions. A classification of simple weight BP k -modules, with finite-dimensional weight spaces, was obtained along with a construction of certain nonsemisimple weight BP k -modules.…”

“…They may be characterised [4] as the subregular (or minimal) quantum hamiltonian reductions of the level-k universal affine vertex algebras V k ( 3 ). This paper is a sequel to [1] in which the representation theory of BP k and its simple quotient BP k was investigated. Here, we are interested in the characters, modular transformations and fusion rules of the simple quotient BP k when k is a nondegenerate admissible level.…”