arXiv subjects

Ozren Perse

Publications and source records attributed to Ozren Perse.

17 recordsLinked to original sources

arXiv2024

A method for describing the maximal ideal in universal affine vertex algebras at non-admissible levels

The problem of determining maximal ideals in universal affine vertex algebras is difficult for levels beyond admissible, since there are no simple character formulas which can be applied. Here we investigate when certain quotient $\mathcal V$ of universal affine vertex algebra $V^k(\mathfrak{g})$ is simple. We present a new method for proving simplicity of quotients of universal affine vertex algebras in the case of affine vertex algebra $L_{k_n}(\mathfrak{sl}_{2n})$ at level $k_n:=-\frac{2n+1}{2}$. In that way we describe the maximal ideal in $V^{k_n}(\mathfrak{sl}_{2n})$. For that purpose, we use the representation theory of minimal affine $W$-algebra $W^{min}_{k_{n+1}}(\mathfrak{sl}_{2n+2})$ developed in [2]. In particular, we use the embedding $L_{k_n}(\mathfrak{sl}_{2n}) \subset W^{min}_{k_{n+1}}(\mathfrak{sl}_{2n+2})$ and fusion rules for $L_{k_n}(\mathfrak{sl}_{2n})$--modules. We apply this result in the cases $n=3,4$ and prove that a maximal ideal is generated by one singular vector of conformal weight $4$. As a byproduct, we classify irreducible modules in the category $\mathcal{O}$ for the simple affine vertex algebra $L_{-7/2}(\mathfrak{sl}_{6})$.

math.QA↗

arXiv2022

Tensor category $KL_k(\mathfrak{sl}_{2n})$ via minimal affine $W$-algebras at the non-admissible level $k =-\frac{2n+1}{2}$

We prove that $KL_k(\mathfrak{sl}_m)$ is a semi-simple, rigid braided tensor category for all even $m\ge 4$, and $k= -\frac{m+1}{2}$ which generalizes result from arXiv:2103.02985 obtained for $m=4$. Moreover, all modules in $KL_k(\mathfrak{sl}_m)$ are simple-currents and they appear in the decomposition of conformal embeddings $\mathfrak{gl}_m \hookrightarrow \mathfrak{sl}_{m+1} $ at level $ k= - \frac{m+1}{2}$ from arXiv:1509.06512. For this we inductively identify minimal affine $W$-algebra $ W_{k-1} (\mathfrak{sl}_{m+2}, θ)$ as simple current extension of $L_{k}(\mathfrak{sl}_m) \otimes \mathcal H \otimes \mathcal M$, where $\mathcal H$ is the rank one Heisenberg vertex algebra, and $\mathcal M$ the singlet vertex algebra for $c=-2$. The proof uses previously obtained results for the tensor categories of singlet algebra from arXiv:2202.05496. We also classify all irreducible ordinary modules for $ W_{k-1} (\mathfrak{sl}_{m+2}, θ)$. The semi-simple part of the category of $ W_{k-1} (\mathfrak{sl}_{m+2}, θ)$-modules comes from $KL_{k-1}(\mathfrak{sl}_{m+2})$, using quantum Hamiltonian reduction, but this $W$-algebra also contains indecomposable ordinary modules.

math.QA↗

arXiv2021

On the representation theory of the vertex algebra $L_{-5/2}(sl(4))$

We study the representation theory of non-admissible simple affine vertex algebra $L_{-5/2} (sl(4))$. We determine an explicit formula for the singular vector of conformal weight four in the universal affine vertex algebra $V^{-5/2} (sl(4))$, and show that it generates the maximal ideal in $V^{-5/2} (sl(4))$. We classify irreducible $L_{-5/2} (sl(4))$--modules in the category ${\mathcal O}$, and determine the fusion rules between irreducible modules in the category of ordinary modules $KL_{-5/2}$. It turns out that this fusion algebra is isomorphic to the fusion algebra of $KL_{-1}$. We also prove that $KL_{-5/2}$ is a semi-simple, rigid braided tensor category. In our proofs we use the notion of collapsing level for the affine $\mathcal{W}$--algebra, and the properties of conformal embedding $gl(4) \hookrightarrow sl(5)$ at level $k=-5/2$ from arXiv:1509.06512. We show that $k=-5/2$ is a collapsing level with respect to the subregular nilpotent element $f_{subreg}$, meaning that the simple quotient of the affine $\mathcal{W}$--algebra $W^{-5/2}(sl(4), f_{subreg})$ is isomorphic to the Heisenberg vertex algebra $M_J(1)$. We prove certain results on vanishing and non-vanishing of cohomology for the quantum Hamiltonian reduction functor $H_{f_{subreg}}$. It turns out that the properties of $H_{f_{subreg}}$ are more subtle than in the case of minimal reducition.

math.QA↗

arXiv2018

Kostant's pair of Lie type and conformal embeddings

We deal with some aspects of the theory of conformal embeddings of affine vertex algebras, providing a new proof of the Symmetric Space Theorem and a criterion for conformal embeddings of equal rank subalgebras. We finally study some examples of embeddings at the critical level. We prove a criterion for embeddings at the critical level which enables us to prove equality of certain central elements.

math.RT↗

arXiv2018

An application of collapsing levels to the representation theory of affine vertex algebras

We discover a large class of simple affine vertex algebras $V_{k} (\mathfrak g)$, associated to basic Lie superalgebras $\mathfrak g$ at non-admissible collapsing levels $k$, having exactly one irreducible $\mathfrak g$-locally finite module in the category ${\mathcal O}$. In the case when $\mathfrak g$ is a Lie algebra, we prove a complete reducibility result for $V_k(\mathfrak g)$-modules at an arbitrary collapsing level. We also determine the generators of the maximal ideal in the universal affine vertex algebra $V^k (\mathfrak g)$ at certain negative integer levels. Considering some conformal embeddings in the simple affine vertex algebras $V_{-1/2} (C_n)$ and $V_{-4}(E_7)$, we surprisingly obtain the realization of non-simple affine vertex algebras of types $B$ and $D$ having exactly one non-trivial ideal.

math.RT↗

arXiv2017

On classification of non-equal rank affine conformal embeddings and applications

We complete the classification of conformal embeddings of a maximally reductive subalgebra $\mathfrak k$ into a simple Lie algebra $\mathfrak g$ at non-integrable non-critical levels $k$ by dealing with the case when $\mathfrak k$ has rank less than that of $\mathfrak g$. We describe some remarkable instances of decomposition of the vertex algebra $V_{k}(\mathfrak g)$ as a module for the vertex subalgebra generated by $\mathfrak k$. We discuss decompositions of conformal embeddings and constructions of new affine Howe dual pairs at negative levels. In particular, we study an example of conformal embeddings $A_1 \times A_1 \hookrightarrow C_3$ at level $k=-1/2$, and obtain explicit branching rules by applying certain $q$-series identity. In the analysis of conformal embedding $A_1 \times D_4 \hookrightarrow C_8$ at level $k=-1/2$ we detect subsingular vectors which do not appear in the branching rules of the classical Howe dual pairs.

math.RT↗

arXiv2017

Conformal embeddings of affine vertex algebras in minimal $W$-algebras II: decompositions

We present methods for computing the explicit decomposition of the minimal simple affine $W$-algebra $W_k(\mathfrak g, θ)$ at a conformal level $k$ as a module for its maximal affine subalgebra $\mathcal V_k(\mathfrak g^{\natural})$. A particular emphasis is given on the application of affine fusion rules to the determination of branching rules. In almost all cases when $\mathfrak g^{\natural}$ is a semisimple Lie algebra, we show that, for a suitable conformal level $k$, $W_k(\mathfrak g, θ)$ is isomorphic to an extension of $\mathcal V_k(\mathfrak g^{\natural})$ by its simple module. We are able to prove that in certain cases $W_k(\mathfrak g, θ)$ is a simple current extension of $\mathcal V_k(\mathfrak g^{\natural})$. In order to analyze more complicated non simple current extensions at conformal levels, we present an explicit realization of the simple $W$-algebra $W_{k}(sl(4), θ)$ at $k=-8/3$. We prove, as conjectured in arXiv:1407.1527, that $W_{k}(sl(4), θ)$ is isomorphic to the vertex algebra $\mathcal R^{(3)}$, and construct infinitely many singular vectors using screening operators. We also construct a new family of simple current modules for the vertex algebra $V_k (sl(n))$ at certain admissible levels and for $V_k (sl(m | n)), m\ne n, m,n\geq 1$ at arbitrary levels.

math.RT↗

arXiv2016

Conformal embeddings of affine vertex algebras in minimal $W$-algebras I: structural results

We find all values of $k\in \mathbb C$, for which the embedding of the maximal affine vertex algebra in a simple minimal W-algebra $W_k(\mathfrak g,θ)$ is conformal, where $\mathfrak g$ is a basic simple Lie superalgebra and $-θ$ its minimal root. In particular, it turns out that if $W_k(\mathfrak g,θ)$ does not collapse to its affine part, then the possible values of these $k$ are either $-\frac{2}{3} h^\vee$ or $-\frac{h^\vee-1}{2}$, where $h^\vee$ is the dual Coxeter number of $\mathfrak g$ for the normalization $(θ,θ)=2$. As an application of our results, we present a realization of simple affine vertex algebra $V_{-\tfrac{n+1}{2} } (sl(n+1))$ inside of the tensor product of the vertex algebra $W_{\tfrac{n-1}{2}} (sl(2| n), θ)$ (also called the Bershadsky-Knizhnik algebra) with a lattice vertex algebra.

math.RT↗

arXiv2016

Finite vs infinite decompositions in conformal embeddings

Building on work of the first and last author, we prove that an embedding of simple affine vertex algebras $V_{\mathbf{k}}(\mathfrak g^0)\subset V_{k}(\mathfrak g)$, corresponding to an embedding of a maximal equal rank reductive subalgebra $\mathfrak g^0$ into a simple Lie algebra $\mathfrak g$, is conformal if and only if the corresponding central charges are equal. We classify the equal rank conformal embeddings. Furthermore we describe, in almost all cases, when $V_{k}(\mathfrak g)$ decomposes finitely as a $V_{\mathbf{k}}(\mathfrak g^0)$-module.

math.RT↗

arXiv2013

Fusion rules and complete reducibility of certain modules for affine Lie algebras

We develop a new method for obtaining branching rules for affine Kac-Moody Lie algebras at negative integer levels. This method uses fusion rules for vertex operator algebras of affine type. We prove that an infinite family of ordinary modules for affine vertex algebra of type A investigated in Adamović and O. Perše (2008) is closed under fusion. Then we apply these fusion rules on explicit bosonic realization of level -1 modules for the affine Lie algebra of type $A_{\ell-1}^{(1)}$, obtain a new proof of complete reducibility for these representations, and the corresponding decomposition for $\ell \ge 3$. We also obtain the complete reducibility of the associated level -1 modules for affine Lie algebra of type $C_{\ell}^{(1)}$. Next we notice that the category of $D_{2 \ell -1}^{(1)}$ modules at level $- 2 \ell +3 $ obtained in Perše (2012) has the isomorphic fusion algebra. This enables us to decompose certain $E_6 ^{(1)}$ and $F_4 ^{(1)}$--modules at negative levels.

math.QA↗

arXiv2012

A note on representations of some affine vertex algebras of type D

In this note we construct a series of singular vectors in universal affine vertex operator algebras associated to $D_{\ell}^{(1)}$ of levels $n-\ell+1$, for $n \in \Z_{>0}$. For $n=1$, we study the representation theory of the quotient vertex operator algebra modulo the ideal generated by that singular vector. In the case $\ell =4$, we show that the adjoint module is the unique irreducible ordinary module for simple vertex operator algebra $L_{D_{4}}(-2,0)$. We also show that the maximal ideal in associated universal affine vertex algebra is generated by three singular vectors.

math.QA↗

arXiv2011

Some general results on conformal embeddings of affine vertex operator algebras

We give a general criterion for conformal embeddings of vertex operator algebras associated to affine Lie algebras at arbitrary levels. Using that criterion, we construct new conformal embeddings at admissible rational and negative integer levels. In particular, we construct all remaining conformal embeddings associated to automorphisms of Dynkin diagrams of simple Lie algebras. The semisimplicity of the corresponding decompositions is obtained by using the concept of fusion rules for vertex operator algebras.

math.QA↗

arXiv2009

Embeddings of vertex operator algebras associated to orthogonal affine Lie algebras

Let $L_{D_{\ell}}(-\ell +{3/2},0)$ (resp. $L_{B_{\ell}}(-\ell +{3/2},0)$) be the simple vertex operator algebra associated to affine Lie algebra of type $D_{\ell}^{(1)}$ (resp. $B_{\ell}^{(1)}$) with the lowest admissible half-integer level $-\ell + {3/2}$. We show that $L_{D_{\ell}}(-\ell +{3/2},0)$ is a vertex subalgebra of $L_{B_{\ell}}(-\ell +{3/2},0)$ with the same conformal vector. For $\ell =4$, $L_{D_{4}}(-{5/2},0)$ is a vertex subalgebra of three copies of $L_{B_{4}}(-{5/2},0)$ contained in $L_{F_{4}}(-{5/2},0)$, and all five of these vertex operator algebras have the same conformal vector.

math.QA↗

arXiv2007

Representations of certain non-rational vertex operator algebras of affine type

In this paper we study a series of vertex operator algebras of integer level associated to the affine Lie algebra $A_{\ell}^{(1)}$. These vertex operator algebras are constructed by using the explicit construction of certain singular vectors in the universal affine vertex operator algebra $N(n-2,0)$ at the integer level. In the case $n=1$ or $l=2$, we explicitly determine Zhu's algebras and classify all irreducible modules in the category $\mathcal{O}$. In the case $l=2$, we show that the vertex operator algebra $N(n-2,0)$ contains two linearly independent singular vectors of the same conformal weight.

math.QA↗

arXiv2007

Vertex operator algebras associated to certain admissible modules for affine Lie algebras of type A

Let $L(-{1/2}(l+1),0)$ be the simple vertex operator algebra associated to an affine Lie algebra of type $A_{l}^{(1)}$ with the lowest admissible half-integer level $-{1/2}(l+1)$, for even l. We study the category of weak modules for that vertex operator algebra which are in category $\cal{O}$ as modules for the associated affine Lie algebra. We classify irreducible objects in that category and prove semisimplicity of that category.

math.QA↗

arXiv2007

Vertex operator algebra analogue of embedding of $B_4$ into $F_4$

Let L_{B}(-5/2,0) (resp. L_{F}(-5/2,0)) be the simple vertex operator algebra associated to affine Lie algebra of type $B_{4}^{(1)}$ (resp. $F_{4}^{(1)}$) with the lowest admissible half-integer level -5/2. We show that L_{B}(-5/2,0) is a vertex subalgebra of L_{F}(-5/2,0) with the same conformal vector, and that L_{F}(-5/2,0) is isomorphic to the extension of L_{B}(-5/2,0) by its only irreducible module other than itself. We also study the representation theory of L_{F}(-5/2,0), and determine the decompositions of irreducible weak L_{F}(-5/2,0)-modules from the category $\mathcal{O}$ into direct sums of irreducible weak L_{B}(-5/2,0)-modules.

math.QA↗

arXiv2006

Vertex operator algebras associated to type B affine Lie algebras on admissible half-integer levels

Let L(n-l+1/2,0) be the vertex operator algebra associated to an affine Lie algebra of type B_l^(1) at level n-l+1/2, for a positive integer n. We classify irreducible L(n-l+1/2,0)-modules and show that every L(n-l+1/2,0)-module is completely reducible. In the special case n=1, we study a category of weak L(-l+3/2,0)-modules which are in the category $\cal{O}$ as modules for the associated affine Lie algebra. We classify irreducible objects in that category and prove semisimplicity of that category.

math.QA↗