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This paper is a natural continuation of our previous work on conformal embeddings of vertex algebras [6], [7], [8]. Here we consider conformal embeddings in simple affine vertex superalgebra Vk(g) where g=g0¯⊕g1¯ is a basic classical simple Lie superalgebra. Let Vk(g0¯) be the subalgebra of Vk(g) generated by g0¯. We first classify all levels k for which the embedding Vk(g0¯) in Vk(g) is conformal. Next we prove that, for a large family of such conformal levels, Vk(g) is a completely reducible Vk(g0¯)–module and obtain decomposition rules. Proofs are based on fusion rules arguments and on the representation theory of certain affine vertex algebras. The most interesting case is the decomposition of V−2(osp(2n+8|2n)) as a finite, non simple current extension of V−2(Dn+4)⊗V1(Cn). This decomposition uses our previous work [10] on the representation theory of V−2(Dn+4).
We also study conformal embeddings gl(n|m)↪sl(n+1|m) and in most cases we obtain decomposition rules.

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... Using this result, and the fact that at level k = −5/4 there is a conformal embedding of L k (sl(2)) into L k (osp(1|2)) (cf. [7], [19,Sect. 10]), we obtain a realization of the Bershadsky-Polyakov algebra W 1 (cf. ...

... For level k = −5/4, there exists a conformal embedding of L k (sl(2)) into L k (osp(1|2)) (cf. [7]), that is, ...

... Set k = −5/4 as before. Using the following decomposition of the conformal embedding from [7], ...

We study the simple Bershadsky–Polyakov algebra 𝒲k = 𝒲k(sl3, fθ) at positive integer levels and classify their irreducible modules. In this way, we confirm the conjecture from [9]. Next, we study the case k = 1. We discover that this vertex algebra has a Kazama–Suzuki-type dual isomorphic to the simple affine vertex superalgebra Lk′ (osp(1|2)) for k′ = –5=4. Using the free-field realization of Lk′ (osp(1|2)) from [3], we get a free-field realization of 𝒲k and their highest weight modules. In a sequel, we plan to study fusion rules for 𝒲k.

... (1) The affine vertex algebras at negative levels which appeared in the decompositions of conformal embeddings in [2], [4], [6]. (2) The affine vertex algebras at collapsing levels for affine W-algebras. ...

... So KL −5/2 is semi-simple. 6. Singular vectors in V k (nω 1 ) and V k (nω 3 ) and the proof of Theorem 5.4 ...

... Straightforward calculation. Since the condition λ i (α ∨ 0 ) / ∈ Z ≥0 is satisfied for i = 1, 6,7,8,9,10,11,16, the assertion (1) gives the singular vectors in these cases. The remaining cases are given by assertion (2). ...

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 this case is much more complicated than the case of minimal reduction.

... This paper is a natural continuation of our previous work on conformal embeddings of vertex algebras [2,3,4,5,7]. In [3] we initiated the study of conformal embeddings of affine vertex algebras in minimal affine W -algebras. ...

... In general, finding these decompositions is a hard problem, open in most cases. Some decompositions are provided in our previous papers [2,4,5,7]. In particular, the paper [5] shows decompositions of minimal affine W -algebras W k (g, x, f θ ). ...

In this paper we prove a general result saying that under mild hypothesis an embedding of an affine vertex algebra into an affine $W$-algebra is conformal if and only if their central charges coincide. This result extends our previous result obtained in the case of minimal affine $W$-algebras. We also find a sufficient condition showing that certain conformal levels are collapsing. This new condition enables us to show collapsing of $W_k(sl(N), x, f )$ for $f$ of hook or rectangular type. Our methods can be applied to non-admissible levels. In particular, we prove Creutzig's conjecture on the conformal embedding in the hook type $W$-algebra $W_k(sl(n+m), x, f_{m,n})$ of its affine vertex subalgebra. Quite surprisingly, the problem of showing that certain conformal levels are not collapsing turns out to be very difficult. In the cases when $k$ is admissible and conformal, we prove that $W_k(sl(n+m), x, f_{m,n})$ is not collapsing. Then, by generalizing the results on semi-simplicity of conformal embeddings from our previous papers, we find many cases in which $W_k(sl(n+m), x, f_{m,n})$ is semi-simple as a module for its affine subalgebra at conformal level and we provide explicit decompositions.

... As outlined before, our present work was made possible by various quite recent results of our research teams. These are a good understanding of conformal embeddings [7,8,10,11,19], collapsing levels [9,18,26], its implications on vertex tensor categories [49] and finally a full understanding of the representation theory of the singlet algebras [40,41]. ...

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}, \theta)$ 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}, \theta)$. The semi-simple part of the category of $ W_{k-1} (\mathfrak{sl}_{m+2}, \theta)$-modules comes from $KL_{k-1}(\mathfrak{sl}_{m+2})$, using quantum Hamiltonian reduction, but this $W$-algebra also contains indecomposable ordinary modules.

... 4.2); (2) Positive rational but generic levels (Sect. 4.4); (3) The levels k such that the ordinary modules for W k (g, θ) are all of finite length (Sect. 4.1). ...

We show that if V is a vertex operator algebra such that all the irreducible ordinary V-modules are \(C_1\)-cofinite and all the grading-restricted generalized Verma modules for V are of finite length, then the category of finite length generalized V-modules has a braided tensor category structure. By applying the general theorem to the simple affine vertex operator algebra (resp. superalgebra) associated to a finite simple Lie algebra (resp. Lie superalgebra) \(\mathfrak {g}\) at level k and the category \(KL_k(\mathfrak {g})\) of its finite length generalized modules, we discover several families of \(KL_k(\mathfrak {g})\) at non-admissible levels k, having braided tensor category structures. In particular, \(KL_k(\mathfrak {g})\) has a braided tensor category structure if the category of ordinary modules is semisimple or more generally if the category of ordinary modules is of finite length. We also prove the rigidity and determine the fusion rules of some categories \(KL_k(\mathfrak {g})\), including the category \(KL_{-1}(\mathfrak {sl}_n)\). Using these results, we construct a rigid tensor category structure on a full subcategory of \(KL_1(\mathfrak {sl}(n|m))\) consisting of objects with semisimple Cartan subalgebra actions.

... We have the following conjecture (which is also in agreement with [24]): Conjecture 8.1. For every even n ≥ 0, we have 10 . ...

We consider several vertex operator algebras and superalgebras closely related to V−1sln, n ≥ 3 : (a) the parafermionic subalgebra K(sl(n); −1) for which we completely describe its inner structure, (b) the vacuum algebra Ω(V−1(sl(n))), and (c) an infinite extension U of V−1(sl(n)) obtained from certain irreducible ordinary modules with integral conformal weights. It turns out that U is isomorphic to the coset vertex algebra psl(n|n)1/sl(n)1, n ≥ 3. We show that V−1(sl(n)) admits precisely n ordinary irreducible modules, up to isomorphism. This leads to the conjecture that U is quasi-lisse.We present evidence in support of this conjecture: we prove that the (super)character of U is quasimodular of weight one by virtue of being the constant term of a meromorphic Jacobi form of index zero. Explicit formulas and MLDE for characters and supercharacters are given for g = sl(3) and outlined for general n. We present a conjectural family of 2nd order MLDEs for characters of vertex algebras psl(n|n)1, n ≥ 2. We finish with a theorem pertaining to characters of psl(n|n)1 and U-modules.

We study the semisimplicity of the category KLk for affine Lie superalgebras and provide a super analog of certain results from [8]. Let KLkfin be the subcategory of KLk consisting of ordinary modules on which a Cartan subalgebra acts semisimply. We prove that KLkfin is semisimple when 1) k is a collapsing level, 2) Wk(g,θ) is rational, 3) Wk(g,θ) is semisimple in a certain category. The analysis of the semisimplicity of KLk is subtler than in the Lie algebra case, since in super case KLk can contain indecomposable modules. We are able to prove that in many cases when KLkfin is semisimple we indeed have KLkfin=KLk, which therefore excludes indecomposable and logarithmic modules in KLk. In these cases we are able to prove that there is a conformal embedding W↪Vk(g) with W semisimple (see Section 10). In particular, we prove the semisimplicity of KLk for g=sl(2|1) and k=−m+1m+2, m∈Z≥0. For g=sl(m|1), we prove that KLk is semisimple for k=−1, but for k a positive integer we show that it is not semisimple by constructing indecomposable highest weight modules in KLkfin.

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.

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.

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.

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 our previous paper J. Algebra319 (2008) 2434–2450, 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 , obtain a new proof of complete reducibility for these representations, and the corresponding decomposition for ℓ ≥ 3. We also obtain the complete reducibility of the associated level -1 modules for affine Lie algebra of type . Next, we notice that the category of modules at level -2ℓ + 3 has the isomorphic fusion algebra. This enables us to decompose certain and -modules at negative levels.

In this paper we investigate the structure of intermediate vertex algebras
associated with a maximal conformal embedding of a reductive Lie algebra in a
semisimple Lie algebra of classical type.

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.
KeywordsVertex operator algebra–Affine Kac–Moody algebra–Conformal embedding–Virasoro algebra–Fusion rules

We investigate vertex operator algebras $L(k,0)$ associated with modular-invariant representations for an affine Lie algebra $A_1 ^{(1)}$ , where k is 'admissible' rational number. We show that VOA $L(k,0)$ is rational in the category $\cal O$ and find all irreducible representations in the category of weight modules.

A field algebra is a ``non-commutative'' generalization of a vertex algebra. In this paper we develop foundations of the theory of field algebras.

The representation theory of affine Kac-Moody Lie algebras has grown tremendously since their independent introduction by Robert V. Moody and Victor G. Kac in 1968. Inspired by mathematical structures found by theoretical physicists, and by the desire to understand the ``monstrous moonshine'' of the Monster group, the theory of vertex operator algebras (VOA's) was introduced by Borcherds, Frenkel, Lepowsky and Meurman. An important subject in this young field is the study of modules for VOA's and intertwining operators between modules. Feingold, Frenkel and Ries defined a structure, called a vertex operator para-algebra(VOPA), where a VOA, its modules and their intertwining operators are unified. In this work, for each $l\geq 1$, we begin with the bosonic construction (from a Weyl algebra) of four level $-\shf$ irreducible representations of the symplectic affine Kac-Moody Lie algebra $C_l^{(1)}$. The direct sum of two of these is given the structure of a VOA, and the direct sum of the other two is given the structure of a twisted VOA-module. In order to define intertwining operators so that the whole structure forms a VOPA, it is necessary to separate the four irreducible modules, taking one As the VOA and the others as modules for it. This work includes the bosonic analog of the fermionic construction of a vertex operator superalgebra from the four level 1 irreducible modules of type $D_l^{(1)}$ given by Feingold, Frenkel and Ries. While they only get a VOPA when $l = 4$ using classical triality, the techniques in this work apply to any $l\geq 1$. Comment: 103

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 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.

This paper is a continuation of arXiv:1602.04687. We present methods for computing the explicit decomposition of the minimal simple affine $W$-algebra $W_k(\mathfrak g, \theta)$ 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 $k$, $W_k(\mathfrak g, \theta)$ 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 $W_k(\mathfrak g, \theta)$ 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 $W_{k}(sl(4), \theta)$ at $k=-8/3$. We prove, as conjectured in arXiv:1407.1527, that $W_{k}(sl(4), \theta)$ is isomorphic to the vertex algebra $\mathcal R^{(3)}$, and construct infinitely many singular vectors using screening operators.

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 $W_k(\mathfrak g,\theta)$ is conformal, where $\mathfrak g$ is a basic simple Lie superalgebra and $-\theta$ its minimal root. In particular, it turns out that if $W_k(\mathfrak g,\theta)$ 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 $(\theta,\theta)=2$. 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 $W_{\tfrac{n-1}{2}} (sl(2| n), \theta)$ (also called the Bershadsky-Knizhnik algebra) with a lattice vertex algebra.

On etudie les contraintes de l'invariance modulaire et conforme dans la theorie de la representation des algebres de Kac-Moody affines. On considere les regles de ramification d'une representation integrable de poids le plus eleve d'une algebre affine g par rapport a une sous-algebre affine p. On etudie certaines fonctions generatrices pour ces coefficients de ramification

A general method for constructing the extension of an ordinary Lie algebra A0 to a superalgebra A0⊕A1 is given, once one knows in which representation of A0 the odd generators A1 are. Explicit matrix representations for the superalgebras F(4) and G(3), and for ordinary algebras E8, F4, and G2 are presented.

We study the structure and representations of a family of vertex algebras obtained from affine superalgebras by quantum reduction. As an application, we obtain in a unified way free field realizations and determinant formulas for all superconformal algebras.

We use modular-invariance and conformal-invariance constraints on the highest-weight representations of affine algebras as well as asymptotic behavior of their characters to deduce an algorithm for the decomposition of these representations with respect to conformal subalgebras. We use the algorithm to find explicit decompositions of level-1 representations of all affine exceptional algebras with respect to all their conformal subalgebras. The crucial point of this work is a connection of the decomposition problem with the Frobenius theory of positive matrices, which hints to a possible connection with Markov chains. The problem of computing branching coefficients for conformal subalgebras arises in the string compactifications. We show that a solution to this problem allows one to construct modular-invariant partition functions on a group manifold. An important new general theoretical result of the paper is the uniqueness of the vacuum state in the basic representation of an affine algebra for its conformal subalgebra.

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. Comment: 40 pages, LaTeX; minor changes

We classify integrable irreducible highest weight representations of non-twisted affine Lie superalgebras. We give a free field construction in the level~1 case. The analysis of this construction shows, in particular, that in the simplest case of the $s\ell (2|1)$ level~1 affine superalgebra the characters are expressed in terms of the Appell elliptic function. Our results demonstrate that the representation theory of affine Lie superalgebras is quite different from that of affine Lie algebras. Comment: 54 pages