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Pierluigi Moseneder Frajria

Publications and source records attributed to Pierluigi Moseneder Frajria.

At least 19 recordsLinked to original sources
arXiv2024

An application of collapsing levels to the representation theory of affine vertex algebras

We discover a large class of simple affine vertex algebras $V_{k} (\mathfrak g)$, associated to basic Lie superalgebras $\mathfrak g$ at non-admissible collapsing levels $k$, having exactly one irreducible $\mathfrak g$-locally finite module in the category ${\mathcal O}$. In the case when $\mathfrak g$ is a Lie algebra, we prove a complete reducibility result for $V_k(\mathfrak g)$-modules at an arbitrary collapsing level. We also determine the generators of the maximal ideal in the universal affine vertex algebra $V^k (\mathfrak g)$ at certain negative integer levels. Considering some conformal embeddings in the simple affine vertex algebras $V_{-1/2} (C_n)$ and $V_{-4}(E_7)$, we surprisingly obtain the realization of non-simple affine vertex algebras of types $B$ and $D$ having exactly one non-trivial ideal.

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arXiv2023

New approaches for studying conformal embeddings and collapsing levels for $W$--algebras

In this paper we prove a general result saying that under certain hypothesis an embedding of an affine vertex algebra into an affine $W$--algebra is conformal if and only if their central charges coincide. This result extends our previous result obtained in the case of minimal affine $W$-algebras. We also find a sufficient condition showing that certain conformal levels are collapsing. This new condition enables us to find some levels $k$ where $W_k(sl(N), x, f )$ collapses to its affine part when $f$ is of hook or rectangular type. Our methods can be applied to non-admissible levels. In particular, we prove Creutzig's conjecture on the conformal embedding in the hook type $W$-algebra $W_k(sl(n+m), x, f_{m,n})$ of its affine vertex subalgebra. Quite surprisingly, the problem of showing that certain conformal levels are not collapsing turns out to be very difficult. In the cases when $k$ is admissible and conformal, we prove that $W_k(sl(n+m), x, f_{m,n})$ is not collapsing. Then, by generalizing the results on semi-simplicity of conformal embeddings from our previous papers, we find many cases in which $W_k(sl(n+m), x, f_{m,n})$ is semi-simple as a module for its affine subalgebra at conformal level and we provide explicit decompositions.

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arXiv2022

On the semisimplicity of the category $KL_k$ for affine Lie superalgebras

We study the semisimplicity of the category $KL_k$ for affine Lie superalgebras and provide a super analog of certain results from arXiv:1801.09880. Let $KL_k^{fin}$ be the subcategory of $KL_k$ consisting of ordinary modules on which the Cartan subalgebra acts semisimply. We prove that $KL_k^{fin}$ is semisimple when 1) $k$ is a collapsing level, 2) $W_k(\mathfrak{g}, θ)$ is rational, 3) $W_k(\mathfrak{g}, θ)$ is semisimple in a certain category. The analysis of the semisimplicity of $KL_k$ is subtler than in the Lie algebra case, since in super case $KL_k$ can contain indecomposable modules. We are able to prove that in many cases when $KL_k^{fin}$ is semisimple we indeed have $KL_k^{fin}=KL_k$, which therefore excludes indecomposable and logarithmic modules in $KL_k$. In these cases we are able to prove that there is a conformal embedding $W \hookrightarrow V_k(\mathfrak{g})$ with $W$ semisimple (see Section 10). In particular, we prove the semisimplicity of $KL_k$ for $\mathfrak{g}=sl(2\vert 1)$ and $k = -\frac{m+1}{m+2}$, $m \in {\mathbb Z}_{\ge 0}$. For $\mathfrak{g} =sl(m \vert 1)$, we prove that $KL_k$ is semisimple for $k=-1$, but for $k=1$ we show that it is not semisimple by constructing indecomposable highest weight modules in $KL_k^{fin}$.

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arXiv2020

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 be a Borel subgroup of G. Then B acts with finitely many orbits on the variety N_2 of the nilpotent elements in g whose height is at most 2. We provide a parametrization of the B-orbits in N_2 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.

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arXiv2020

Yangians vs minimal W-algebras: a surprizing coincidence

We prove that the singularities of the $R$-matrix $R(k)$ of the minimal quantization of the adjoint representation of the Yangian $Y(\mathfrak g)$ of a finite dimensional simple Lie algebra $\mathfrak g$ are the opposite of the roots of the monic polynomial $p(k)$ entering in the OPE expansions of quantum fields of conformal weight $3/2$ of the universal minimal affine $W$-algebra at level $k$ attached to $\mathfrak g$.

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arXiv2020

The Bruhat order on abelian ideals of Borel subalgebras

Let G be a quasi simple algebraic group over an algebraically closed field k whose characteristic is not very bad for G, and let B be a Borel subgroup of G with Lie algebra b. Given a B-stable abelian subalgebra a of the nilradical of b, we parametrize the B-orbits in a and we describe their closure relations.

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arXiv2018

Kostant's pair of Lie type and conformal embeddings

We deal with some aspects of the theory of conformal embeddings of affine vertex algebras, providing a new proof of the Symmetric Space Theorem and a criterion for conformal embeddings of equal rank subalgebras. We finally study some examples of embeddings at the critical level. We prove a criterion for embeddings at the critical level which enables us to prove equality of certain central elements.

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arXiv2017

On classification of non-equal rank affine conformal embeddings and applications

We complete the classification of conformal embeddings of a maximally reductive subalgebra $\mathfrak k$ into a simple Lie algebra $\mathfrak g$ at non-integrable non-critical levels $k$ by dealing with the case when $\mathfrak k$ has rank less than that of $\mathfrak g$. We describe some remarkable instances of decomposition of the vertex algebra $V_{k}(\mathfrak g)$ as a module for the vertex subalgebra generated by $\mathfrak k$. We discuss decompositions of conformal embeddings and constructions of new affine Howe dual pairs at negative levels. In particular, we study an example of conformal embeddings $A_1 \times A_1 \hookrightarrow C_3$ at level $k=-1/2$, and obtain explicit branching rules by applying certain $q$-series identity. In the analysis of conformal embedding $A_1 \times D_4 \hookrightarrow C_8$ at level $k=-1/2$ we detect subsingular vectors which do not appear in the branching rules of the classical Howe dual pairs.

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arXiv2017

Conformal embeddings of affine vertex algebras in minimal $W$-algebras II: decompositions

We present methods for computing the explicit decomposition of the minimal simple affine $W$-algebra $W_k(\mathfrak g, θ)$ at a conformal level $k$ as a module for its maximal affine subalgebra $\mathcal V_k(\mathfrak g^{\natural})$. A particular emphasis is given on the application of affine fusion rules to the determination of branching rules. In almost all cases when $\mathfrak g^{\natural}$ is a semisimple Lie algebra, we show that, for a suitable conformal level $k$, $W_k(\mathfrak g, θ)$ is isomorphic to an extension of $\mathcal V_k(\mathfrak g^{\natural})$ by its simple module. We are able to prove that in certain cases $W_k(\mathfrak g, θ)$ is a simple current extension of $\mathcal V_k(\mathfrak g^{\natural})$. In order to analyze more complicated non simple current extensions at conformal levels, we present an explicit realization of the simple $W$-algebra $W_{k}(sl(4), θ)$ at $k=-8/3$. We prove, as conjectured in arXiv:1407.1527, that $W_{k}(sl(4), θ)$ is isomorphic to the vertex algebra $\mathcal R^{(3)}$, and construct infinitely many singular vectors using screening operators. We also construct a new family of simple current modules for the vertex algebra $V_k (sl(n))$ at certain admissible levels and for $V_k (sl(m | n)), m\ne n, m,n\geq 1$ at arbitrary levels.

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arXiv2016

Spherical nilpotent orbits and abelian subalgebras in isotropy representations

Let $G$ be a simply connected semisimple algebraic group with Lie algebra $\mathfrak g$, let $G_0 \subset G$ be the symmetric subgroup defined by an algebraic involution $σ$ and let $\mathfrak g_1 \subset \mathfrak g$ be the isotropy representation of $G_0$. Given an abelian subalgebra $\mathfrak a$ of $\mathfrak g$ contained in $\mathfrak g_1$ and stable under the action of some Borel subgroup $B_0 \subset G_0$, we classify the $B_0$-orbits in $\mathfrak a$ and we characterize the sphericity of $G_0 \mathfrak a$. Our main tool is the combinatorics of $σ$-minuscule elements in the affine Weyl group of $\mathfrak g$ and that of strongly orthogonal roots in Hermitian symmetric spaces.

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arXiv2016

Conformal embeddings of affine vertex algebras in minimal $W$-algebras I: structural results

We find all values of $k\in \mathbb C$, for which the embedding of the maximal affine vertex algebra in a simple minimal W-algebra $W_k(\mathfrak g,θ)$ is conformal, where $\mathfrak g$ is a basic simple Lie superalgebra and $-θ$ its minimal root. In particular, it turns out that if $W_k(\mathfrak g,θ)$ does not collapse to its affine part, then the possible values of these $k$ are either $-\frac{2}{3} h^\vee$ or $-\frac{h^\vee-1}{2}$, where $h^\vee$ is the dual Coxeter number of $\mathfrak g$ for the normalization $(θ,θ)=2$. As an application of our results, we present a realization of simple affine vertex algebra $V_{-\tfrac{n+1}{2} } (sl(n+1))$ inside of the tensor product of the vertex algebra $W_{\tfrac{n-1}{2}} (sl(2| n), θ)$ (also called the Bershadsky-Knizhnik algebra) with a lattice vertex algebra.

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arXiv2016

Finite vs infinite decompositions in conformal embeddings

Building on work of the first and last author, we prove that an embedding of simple affine vertex algebras $V_{\mathbf{k}}(\mathfrak g^0)\subset V_{k}(\mathfrak g)$, corresponding to an embedding of a maximal equal rank reductive subalgebra $\mathfrak g^0$ into a simple Lie algebra $\mathfrak g$, is conformal if and only if the corresponding central charges are equal. We classify the equal rank conformal embeddings. Furthermore we describe, in almost all cases, when $V_{k}(\mathfrak g)$ decomposes finitely as a $V_{\mathbf{k}}(\mathfrak g^0)$-module.

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arXiv2013

Symmetries of abelian ideals of Borel subalgebras

Elaborating on a paper by Suter, we provide a detailed description of the automorphism group of the poset of abelian ideals in a Borel subalgebra of a finite dimensional complex simple Lie algebra.

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arXiv2012

Denominator identities for finite-dimensional Lie superalgebras and Howe duality for compact dual pairs

We provide formulas for the denominator and superdenominator of a basic classical type Lie superalgebra for any set of positive roots. We establish a connection between certain sets of positive roots and the theory of reductive dual pairs of real Lie groups. As an application of our formulas, we recover the Theta correspondence for compact dual pairs. Along the way we give an explicit description of the real forms of basic classical type Lie superalgebras.

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arXiv2009

On the Kernel of the affine Dirac operator

Let L be a finite-dimensional semisimple Lie algebra with a non-degenerate invariant bilinear form, σan elliptic automorphism of L leaving the form invariant, and A a σ-invariant reductive subalgebra of L, such that the restriction of the form to A is non-degenerate. Consider the associated twisted affine Lie algebras L^, A^, and let F be the σ-twisted Clifford module over A^ associated to the orthocomplement of A in L. Under suitable hypotheses onσand A, we provide a general formula for the decomposition of the kernel of the affine Dirac operator, acting on the tensor product of an integrable highest weight L^-module and F, into irreducible A^-submodules. As an application, we derive the decomposition of all level 1 integrable irreducible highest weight modules over orthogonal affine Lie algebras with respect to the affinization of the isotropy subalgebra of an arbitrary symmetric space.

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