Compatible Discrete Series

Several very interesting results connecting the theory of abelian ideals of Borel subalgebras, some ideas of D. Peterson relating the previous theory to the combinatorics of affine Weyl groups, and the theory of discrete series are stated in a recent paper (\cite{Ko2}) by B. Kostant. In this paper we provide proofs for most of Kostant's results extending them to $ad$-nilpotent ideals and develop one direction of Kostant's investigation, the compatible discrete series.Comment: AmsTex file, 27 Pages; minor corrections; to appear in Pacific Journal of Mathematic

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

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

Confronting Patients: Therapists' Model of a Responsiveness Based Approach

Confrontation represents a way of challenging patients in psychotherapy to stimulate change. Confrontation draws attention to discrepancies, for example between elements in a patient’s functioning. The present study was designed to construct a conceptual model of confrontation used by therapists when trying to address two main questions: what are the risks and opportunities of confrontation and how can these effects be influenced? Fifteen therapists from the Psychotherapy Outpatient Clinic of the University of Bern in Switzerland participated in semi-standardized interviews, which were analyzed using qualitative content analysis and thematic analysis. Several main themes merged into a dynamic, sequential model: groundwork required before a confrontation, shaping the confrontation, the (immediate) effects, and management of negative consequences. Therapists assume that a confrontation may induce insight and can strengthen the therapeutic relationship either directly or indirectly through the repair of a rupture in the alliance

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

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 $W$-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