Searcharxiv⌕ Search

arXiv subjects

Anne Moreau

Publications and source records attributed to Anne Moreau.

At least 19 recordsLinked to original sources
arXiv2025

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.

math.QA↗
arXiv2024

On some simple orbifold affine VOAs at non-admissible level arising from rank one 4D SCFTs

We study the representations of some simple affine vertex algebras at non-admissible level arising from rank one 4D SCFTs. In particular, we classify the irreducible highest weight modules of $L_{-2}(G_2)$ and $L_{-2}(B_3)$. It is known by the works of Adamovi\'{c} and Per\v{s}e that these vertex algebras can be conformally embedded into $L_{-2}(D_4)$. We also compute the associated variety of $L_{-2}(G_2)$, and show that it is the orbifold of the associated variety of $L_{-2}(D_4)$ by the symmetric group of degree 3 which is the Dynkin diagram automorphism group of $D_4$. This provides a new interesting example of associated variety satisfying a number of conjectures in the context of orbifold vertex algebras.

math.RT↗
arXiv2023

Functorial constructions related to double Poisson vertex algebras

For any double Poisson algebra, we produce a double Poisson vertex algebra using the jet algebra construction. We show that this construction is compatible with the representation functor which associates to any double Poisson (vertex) algebra and any positive integer a Poisson (vertex) algebra. We also consider related constructions, such as Poisson reductions and Hamiltonian reductions, with the aim of comparing the different corresponding categories. This allows us to provide various interesting examples of double Poisson vertex algebras, in particular from double quivers.

math.RT↗
arXiv2022

Action of the automorphism group on the Jacobian of Klein's quartic curve II: Invariant theta functions

Bernstein-Schwarzman conjectured that the quotient of a complex affine space by an irreducible complex crystallographic group generated by reflections is a weighted projective space. The conjecture was proved by Schwarzman and Tokunaga-Yoshida in dimension 2 for almost all such groups, and for all crystallographic reflection groups of Coxeter type by Looijenga, Bernstein-Schwarzman and Kac-Peterson in any dimension. We prove that the conjecture is true for the crystallographic reflection group in dimension 3 for which the associated collineation group is Klein's simple group of order 168. In this case the quotient is the 3-dimensional weighted projective space with weights 1, 2, 4, 7. The main ingredient in the proof is the computation of the algebra of invariant theta functions. Unlike the Coxeter case, the invariant algebra is not free polynomial, and this was the major stumbling block.

math.AG↗
arXiv2021

Action of the automorphism group on the Jacobian of Klein's quartic curve

Klein's simple group $H$ of order $168$ is the automorphism group of the plane quartic curve $C$, called Klein quartic. By Torelli Theorem, the full automorphism group $G$ of the Jacobian $J=J(C)$ is the group of order $336$, obtained by adding minus identity to $H$. The quotient variety $J/G$ can be alternatively represented as the quotient $\mathbb C^3/\tilde G$ of the complex $3$-space by the complex crystallographic group $\tilde G$, the extension of $G$ by the period lattice of the Klein quartic. Moreover, it turns out that $\tilde G$ is generated by affine complex reflections. According to a conjecture of Bernstein--Schwarzman, a quotient of $\mathbb C^n$ by an irreducible complex crystallographic group generated by reflections is a weighted projective space. The conjecture is known in dimension two and for complexifications of the real crystallographic groups generated by reflections. The case of $\tilde G$ is the first, and in a sense the smallest of the unknown cases. We compute the orbits and the stabilizers of the action of $G$ on $J$ and deduce that $J/G=\mathbf C^3/\tilde G$ is a strongly simply connected variety with the same singularities as the weighted projective space $\mathbb P(1,2,4,7)$.

math.AG↗
arXiv2021

Singularities of nilpotent Slodowy slices and collapsing levels of W-algebras

We apply results from the geometry of nilpotent orbits and nilpotent Slodowy slices, together with modularity and asymptotic analysis of characters, to prove many new isomorphisms between affine W-algebras and affine Kac-Moody vertex algebras and their finite extensions at specific admissible levels. In particular we identify many new collapsing levels for W-algebras.

math.RT↗
arXiv2020

Simplicity of vacuum modules and associated varieties

In this note, we prove that the universal affine vertex algebra associated with a simple Lie algebra $\mathfrak{g}$ is simple if and only if the associated variety of its unique simple quotient is equal to $\mathfrak{g}^*$. We also derive an analogous result for the quantized Drinfeld-Sokolov reduction applied to the universal affine vertex algebra.

math.RT↗
arXiv2020

Kostant principal filtration and paths in weight lattices

There are several interesting filtrations on the Cartan subalgebra of a complex simple Lie algebra coming from very different contexts: one is the principal filtration coming from the Langlands dual, one is coming from the Clifford algebra associated with a non-degenerate invariant bilinear form, one is coming from the symmetric algebra and the Chevalley projection, and two other ones are coming from the enveloping algebra and Harish-Chandra projections. It is now known that all these filtrations coincide. This results from a combination of works of several authors (Rohr, Joseph, Alekseev and the second named author), and was essentially conjectured by Kostant. In this paper, we establish a direct correspondence between the enveloping filtration and the symmetric filtration for a simple Lie algebra of type A or C. Our proof is very different from Rohr and Joseph approaches. The idea is to use an explicit description of the symmetric and enveloping invariants in term of combinatorial objects, called weighted paths, in the crystal graph of the standard representation.

math.RT↗
arXiv2019

Sheets and associated varieties of affine vertex algebras

We show that sheet closures appear as associated varieties of affine vertex algebras. Further, we give new examples of non-admissible affine vertex algebras whose associated variety is contained in the nilpotent cone. We also prove some conjectures from our previous paper and give new examples of lisse affine W-algebras.

math.RT↗
arXiv2019

Arc spaces and chiral symplectic cores

We introduce the notion of chiral symplectic cores in a vertex Poisson variety, which can be viewed as analogs of symplectic leaves in Poisson varieties. As an application we show that any quasi-lisse vertex algebra is a quantization of the arc space of its associated variety, in the sense that its reduced singular support coincides with the reduced arc space of its associated variety. We also show that the coordinate ring of the arc space of Slodowy slices is free over its vertex Poisson center, and the latter coincides with the vertex Poisson center of the coordinate ring of the arc space of the dual of the corresponding simple Lie algebra.

math.RT↗
arXiv2018

Satellites of spherical subgroups

Let $G$ be a complex connected reductive algebraic group. Given a spherical subgroup $H \subset G$ and a subset $I$ of the set of spherical roots of $G/H$, we define, up to conjugation, a spherical subgroup $H_I \subset G$ of the same dimension of $H$, called a satellite. We investigate various interpretations of the satellites. We also show a close relation between the Poincaré polynomials of the two spherical homogeneous spaces $G/H$ and $G/H_I$.

math.AG↗
arXiv2017

A remark on Mishchenko-Fomenko algebras and regular sequences

In this note, we show that the free generators of the Mishchenko-Fomenko subalgebra of a complex reductive Lie algebra, constructed by the argument shift method at a regular element, form a regular sequence. This result was proven by Serge Ovsienko in the type A at a regular and semisimple element. Our approach is very different, and is strongly based on geometric properties of the nilpotent bicone.

math.RT↗
arXiv2016

The symmetric invariants of centralizers and Slodowy grading II

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra of rank $\ell$ over an algebraically closed field $\Bbbk$ of characteristic zero, and let $(e,h,f)$ be an $\mathfrak{sl}_2$-triple of g. Denote by $\mathfrak{g}^{e}$ the centralizer of $e$ in $\mathfrak{g}$ and by ${\rm S}(\mathfrak{g}^{e})^{\mathfrak{g}^{e}}$ the algebra of symmetric invariants of $\mathfrak{g}^{e}$. We say that $e$ is good if the nullvariety of some $\ell$ homogenous elements of ${\rm S}(\mathfrak{g}^{e})^{\mathfrak{g}^{e}}$ in $(\mathfrak{g}^{e})^{*}$ has codimension $\ell$. If $e$ is good then ${\rm S}(\mathfrak{g}^{e})^{\mathfrak{g}^{e}}$ is a polynomial algebra. In this paper, we prove that the converse of the main result of arXiv:1309.6993 is true. Namely, we prove that $e$ is good if and only if for some homogenous generating sequence $q_1,\ldots,q_\ell$, the initial homogenous components of their restrictions to $e+\mathfrak{g}^{f}$ are algebraically independent over $\Bbbk$.

math.RT↗
arXiv2016

On the irreducibility of associated varieties of W-algebras

We investigate the irreducibility of the nilpotent Slodowy slices that appear as the associated variety of W-algebras. Furthermore, we provide new examples of vertex algebras whose associated variety has finitely many symplectic leaves.

math.RT↗
arXiv2016

Joseph ideals and lisse minimal W-algebras

We consider a lifting of Joseph ideals for the minimal nilpotent orbit closure to the setting of affine Kac-Moody algebras and find new examples of affine vertex algebras whose associated varieties are minimal nilpotent orbit closures. As an application we obtain a new family of lisse ($C_2$-cofinite) W-algebras that are not coming from admissible representations of affine Kac-Moody algebras.

math.RT↗
arXiv2015

The index of centralizers of elements of reductive Lie algebras

For a finite dimensional complex Lie algebra, its index is the minimal dimension of stabilizers for the coadjoint action. A famous conjecture due to Elashvili says that the index of the centralizer of an element of a reductive Lie algebra is equal to the rank. That conjecture caught attention of several Lie theorists for years. In this paper we give an almost general proof of that conjecture.

math.RT↗