Ana Kontrec

Bio: Ana Kontrec is an academic researcher from University of Zagreb. The author has contributed to research in topics: Vertex operator algebra & Representation theory. The author has an hindex of 2, co-authored 3 publications receiving 13 citations.

Classification of irreducible modules for Bershadsky–Polyakov algebra at certain levels

Abstract: We study the representation theory of the Bershadsky–Polyakov algebra 𝒲k = 𝒲k(sl3,f𝜃). In particular, Zhu algebra of 𝒲k is isomorphic to a certain quotient of the Smith algebra, after changing the ...

Classification of irreducible modules for Bershadsky-Polyakov algebra at certain levels

Abstract: We study the representation theory of the Bershadsky-Polyakov algebra $\mathcal W_k = \mathcal{W}_k(sl_3,f_{\theta})$. In particular, Zhu algebra of $\mathcal W_k$ is isomorphic to a certain quotient of the Smith algebra, after changing the Virasoro vector. We classify all modules in the category $\mathcal{O}$ for the Bershadsky-Polyakov algebra $\mathcal W_k$ when $k=-5/3, -9/4, -1,0$. In the case $k=0$ we show that the Zhu algebra $A(\mathcal W_k)$ has $2$--dimensional indecomposable modules.

Bershadsky-Polyakov vertex algebras at positive integer levels and duality

Abstract: We study the simple Bershadsky-Polyakov algebra $\mathcal W_k = \mathcal{W}_k(sl_3,f_{\theta})$ at positive integer levels and classify their irreducible modules. In this way we confirm the conjecture from arXiv:1910.13781. Next, we study the case $k=1$. We discover that this vertex algebra has a Kazama-Suzuki-type dual isomorphic to the simple afine vertex superalgebra $L_{k'} (osp(1 \vert 2))$ for $k'=-5/4$. Using the free-field realization of $L_{k'} (osp(1 \vert 2))$ from arXiv:1711.11342, we get a free-field realization of $\mathcal W_k$ and their highest weight modules. In a sequel, we plan to study fusion rules for $\mathcal W_k$.

Multiplets of representations, twisted Dirac operators and Vogan's conjecture in affine setting

Abstract: We extend classical results of Kostant and al. on multiplets of representations of finite-dimensional Lie algebras and on the cubic Dirac operator to the setting of affine Lie algebras and twisted affine cubic Dirac operator. We prove in this setting an analogue of Vogan's conjecture on infinitesimal characters of Harish-Chandra modules in terms of Dirac cohomology. For our calculations we use the machinery of Lie conformal and vertex algebras.

Classifying Relaxed Highest-Weight Modules for Admissible-Level Bershadsky–Polyakov Algebras

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 $$\mathsf {k}+\frac{3}{2} \in \mathbb {Z}_{\geqslant 0}$$ .

A realisation of the Bershadsky–Polyakov algebras and their relaxed modules

Abstract: We present a realisation of the universal/simple Bershadsky–Polyakov vertex algebras as subalgebras of the tensor product of the universal/simple Zamolodchikov vertex algebras and an isotropic lattice vertex algebra. This generalises the realisation of the universal/simple affine vertex algebras associated to $$\mathfrak {sl}_{2}$$ and $$\mathfrak {osp} (1 \vert 2)$$ given in Adamovic (Commun Math Phys 366:1025–1067, 2019). Relaxed highest-weight modules are likewise constructed, conditions for their irreducibility are established, and their characters are explicitly computed, generalising the character formulae of Kawasetsu and Ridout (Commun Math Phys 368:627–663, 2019).

A realisation of the Bershadsky--Polyakov algebras and their relaxed modules

Abstract: We present a realisation of the universal/simple Bershadsky--Polyakov vertex algebras as subalgebras of the tensor product of the universal/simple Zamolodchikov vertex algebras and an isotropic lattice vertex algebra. This generalises the realisation of the universal/simple affine vertex algebras associated to $\mathfrak{sl}_2$ and $\mathfrak{osp}(1|2)$ given in arXiv:1711.11342. Relaxed highest-weight modules are likewise constructed, conditions for their irreducibility are established, and their characters are explicitly computed, generalising the character formulae of arXiv:1803.01989.