Dirac operators and the very strange formula for Lie superalgebras

Using a super-affine version of Kostant’s cubic Dirac operator, we prove a very strange formula for quadratic finite-dimensional Lie superalgebras with a reductive even subalgebra

Conformal embeddings in affine vertex superalgebras

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 $V_k(\mathfrak g)$ where $\mathfrak g=\mathfrak g_{\bar 0}\oplus \mathfrak g_{\bar 1}$ is a basic classical simple Lie superalgebras. Let $\mathcal V_k (\mathfrak g_{\bar 0})$ be the subalgebra of $V_k(\mathfrak g)$ generated by $\mathfrak g_{\bar 0}$. We first classify all levels $k$ for which the embedding $\mathcal V_k (\mathfrak g_{\bar 0})$ in $V_k(\mathfrak g)$ is conformal. Next we prove that, for a large family of such conformal levels, $V_k(\mathfrak g)$ is a completely reducible $\mathcal V_k (\mathfrak g_{\bar 0})$--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 $V_{-2} (osp(2n +8 \vert 2n))$ as a finite, non simple current extension of $V_{-2} (D_{n+4}) \otimes V_1 (C_n)$. This decomposition uses our previous work [10] on the representation theory of $V_{-2} (D_{n+4})$.Comment: Latex file, 45 pages, to appear in Advances in Mathematic

Nilpotent orbits of height 2 and involutions in the affine Weyl group

Let G be an almost simple group over an algebraically closed field k of characteristic zero, let g be its Lie algebra and let B G be a Borel subgroup. Then B acts with finitely many orbits on the variety N2 of the nilpotent elements whose height is at most 2. We provide a parametrization of the B-orbits in N2 in terms of subsets of pairwise orthogonal roots, and we provide a complete description of the inclusion order among the B-orbit closures in terms of the Bruhat order on certain involutions in the affine Weyl group of g

On special covariants in the exterior algebra of a simple Lie algebra

We study the subspace of the exterior algebra of a simple complex Lie algebra linearly spanned by the copies of the little adjoint representation or, in the case of the Lie algebra of traceless matrices, by the copies of the n-th symmetric power of the defining representation. As main result we prove that this subspace is a free module over the subalgebra of the exterior algebra generated by all primitive invariants except the one of highest degree.Comment: Latex file, 11 pages, Final version, appeared in "Rendiconti Lincei - Matematica e Applicazioni

Invariant Hermitian forms on vertex algebras

We study invariant Hermitian forms on a conformal vertex algebra and on their (twisted) modules. We establish existence of a non-zero invariant Hermitian form on an arbitrary $W$-algebra. We show that for a minimal simple $W$-algebra $W_k(\mathfrak g,\theta/2)$ this form can be unitary only when its $\tfrac{1}{2}\mathbb Z$-grading is compatible with parity, unless $W_k(\mathfrak g,\theta/2)$ ''collapses'' to its affine subalgebra.Comment: Latex file, 33 page

Conformal Embeddings and Simple Current Extensions

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

Unitarity of minimal WW-algebras

We obtain a complete classification of minimal simple unitary $W$-algebras.Comment: Latex file, 18 page

Unitarity of minimal WW-algebras and their representations I

We begin a systematic study of unitary representations of minimal $W$-algebras. In particular, we classify unitary minimal $W$-algebras and make substantial progress in classification of their unitary irreducible highest weight modules. We also compute the characters of these modules.Comment: Latex file, 60 pages. arXiv admin note: text overlap with arXiv:2012.1464