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Pierluigi Möseneder Frajria

Publications and source records attributed to Pierluigi Möseneder Frajria.

11 recordsLinked to original sources
arXiv2026

W-algebras as conformal extensions of affine VOAs

We provide a criterion for a vertex operator superalgebra homomorphism from an affine vertex algebra to another vertex superalgebra to be conformal, and an additional criterion that guarantees that this homomorphism is surjective. This situation is applied to W-algebras and W-superalgebras and we list all cases where our criterion applies. This gives many new examples of W-algebras that collapse to affine vertex algebras or are conformal extensions. In particular, we provide many examples of simple W-algebras at non-admissible levels that collapse to admissible level affine vertex algebras.

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arXiv2026

Spectral flow and application to unitarity of representations of minimal $W$-algebras

Using spectral flow, we provide a proof of [11, Theorem 9.17] on unitarity of Ramond twisted non-extremal representations of unitary minimal $W$-algebras that does not rely on the still conjectural exactness of the twisted quantum reduction functor (see Conjecture 9.11 of [11]). When $\mathfrak g = spo(2|2n)$, $F (4$), $D(2, 1; \frac{m}{n})$, it is also proven that the unitarity of extremal (=massless) representations of the unitary minimal $W$-algebra $W^k_{\min}(\mathfrak g)$ in the Ramond sector is equivalent to the unitarity of extremal representations in the Neveu-Schwarz sector.

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arXiv2026

Extremal unitary representations of big $N=4$ superconformal algebra

In this paper we give a detailed proof of the classification of extremal (=massless) unitary highest weight representations in the Neveu Schwarz and Ramond sectors of the big $N=4$ superconformal algebra which can be found in [5]. Our results agree with the general conjectures about classification of unitary highest weight representation of minimal $W$-algebras attached to basic Lie superalgebras formulated in [10], [11], and complete their proof for the big $N=4$ superconformal algebra.

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arXiv2025

Unitarity of minimal $W$-algebras and their representations II: Ramond sector

In this paper we study unitary Ramond twisted representations of minimal $W$-algebras. We classify all such irreducible highest weight representations with a non-Ramond extremal highest weight (unitarity in the Ramond extremal case, as well as in the untwisted extremal case, remains open). We compute the characters of these representations and deduce from them the denominator identities for all superconformal algebras in the Neveu-Schwarz and Ramond sector. Some of the results rely on conjectures about the properties of the quantum Hamiltonian reduction functor in the Ramond sector.

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arXiv2023

Defining relations for minimal unitary quantum affine W-algebras

We prove that any unitary highest weight module over a universal minimal quantum affine $W$-algebra at non-critical level descends to its simple quotient. We find the defining relations of the unitary simple minimal quantum affine $W$-algebras and the list of all their irreducible positive energy modules. We also classify all irreducible highest weight modules for the simple affine vertex algebras in the cases when the associated simple minimal $W$-algebra is unitary.

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arXiv2023

Unitarity of minimal $W$-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.

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arXiv2021

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 $W_k(\mathfrak g,θ/2)$ this form can be unitary only when its $\tfrac{1}{2}\mathbb Z$-grading is compatible with parity, unless $W_k(\mathfrak g,θ/2)$ "collapses" to its affine subalgebra.

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arXiv2019

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 $V_{-2} (D_{n+4})$.

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arXiv2014

The maximum cardinality of minimal inversion complete sets in finite reflection groups

We compute for reflection groups of type $A,B,D,F_4,H_3$ and for dihedral groups a statistic counting the maximal cardinality of a set of elements in the group whose generalized inversions yield the full set of inversions and which are minimal with respect to this property. We also provide lower bounds for the $E$ types that we conjecture to be the exact value of our statistic.

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arXiv2014

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.

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