51 research outputs found
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 where
is a basic
classical simple Lie superalgebras. Let
be the subalgebra of generated by . We
first classify all levels for which the embedding in is conformal. Next we prove that, for a
large family of such conformal levels, is a completely
reducible --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 as a finite, non
simple current extension of . This
decomposition uses our previous work [10] on the representation theory of
.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
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 -algebra. We show that for a minimal simple -algebra
this form can be unitary only when its
-grading is compatible with parity, unless
''collapses'' to its affine subalgebra.Comment: Latex file, 33 page
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
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 -algebras
We obtain a complete classification of minimal simple unitary -algebras.Comment: Latex file, 18 page
Unitarity of minimal -algebras and their representations I
We begin a systematic study of unitary representations of minimal
-algebras. In particular, we classify unitary minimal -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
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