Classifying Relaxed Highest-Weight Modules for Admissible-Level Bershadsky–Polyakov Algebras
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In this article, Adamovic and Kontrec showed that the simple Bershadsky-polyakov algebras with admissible non-integral weights are always rational in the category of highest-weight modules.Abstract:
The Bershadsky–Polyakov algebras are the minimal quantum hamiltonian reductions of the affine vertex algebras associated to $$\mathfrak {sl}_{3}$$
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, with finite-dimensional weight spaces, for all admissible but nonintegral levels, significantly generalising the known highest-weight classifications (Arakawa in Commun Math Phys 323:627–633, 2013, Adamovic and Kontrec in Classification of irreducible modules for Bershadsky–Polyakov algebra at certain levels). In particular, we prove that the simple Bershadsky–Polyakov algebras with admissible nonintegral $$\mathsf {k}$$
are always rational in category $$\mathscr {O}$$
, whilst they always admit nonsemisimple relaxed highest-weight modules unless $$\mathsf {k}+\frac{3}{2} \in \mathbb {Z}_{\geqslant 0}$$
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Citations
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A realisation of the Bershadsky–Polyakov algebras and their relaxed modules
TL;DR: In this paper, a realisation of the universal/simple Bershadsky-Polyakov vertex algebra as subalgebras of the tensor product of the Zamolodchikov vertex algebra and an isotropic lattice vertex algebra is presented.
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Simple modules for Affine vertex algebras in the minimal nilpotent orbit
TL;DR: In this paper, a new family of simple positive energy representations for the simple affine vertex algebra V_k(sl n+1) in the minimal nilpotent orbit of sl n + 1 has been constructed in terms of Gelfand-Tsetlin tableaux.
Journal ArticleDOI
A realisation of the Bershadsky--Polyakov algebras and their relaxed modules
TL;DR: In this paper, a realisation of the universal/simple Bershadsky-Polyakov vertex algebra as subalgebras of the tensor product of the Zamolodchikov vertex algebra and an isotropic lattice vertex algebra is presented.
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Modularity of Bershadsky–Polyakov minimal models
TL;DR: In this article , Adamović et al. studied the modular properties of Bershadsky-Polyakov characters and deduced the associated Grothendieck fusion rules.
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Bershadsky-Polyakov vertex algebras at positive integer levels and duality
Drazen Adamovic,Ana Kontrec +1 more
TL;DR: In this article, the authors study the simple Bershadsky-polyakov algebra W_k = \mathcal{W}_k(sl_3,f_{\theta})$ at positive integer levels and classify their irreducible modules.
References
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Posted Content
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