arXiv subjects

Florence Fauquant-Millet

Publications and source records attributed to Florence Fauquant-Millet.

9 recordsLinked to original sources

arXiv2026

Polynomiality for algebras of invariants associated with some In\"on\"u-Wigner contractions in type A

This paper deals with polynomiality of algebras of symmetric invariants (or generated by symmetric semi-invariants) associated with some particular In\"on\"u-Wigner contractions in type A. More precisely we are interested with contractions of some parabolic subalgebras in type A with respect to their decomposition into their Levi factor and their nilpotent radical, the latter becoming an abelian ideal of the contraction. When the parabolic subalgebra $\mathfrak p$ has its Levi factor decomposing into three symmetric blocks, with respect to the antidiagonal, we show in this article that the algebra of symmetric invariants associated with the contraction of the canonical truncation of $\mathfrak p$ is a polynomial algebra, for which we can give the number of algebraically independent homogeneous generators, their weight and degree. The method relies on the construction of an adapted pair for such a contraction. As a by-product we obtain that the algebra generated by symmetric semi-invariants associated with the In\"on\"u-Wigner contraction of such a parabolic subalgebra $\mathfrak p$ in type A has a Weierstrass section and then is a polynomial algebra, when the central block is not of the same size as the two extremal blocks of the Levi factor.

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arXiv2025

Symmetric semi-invariants for some Inonu-Wigner contractions-II-Case B even

We consider a proper parabolic subalgebra p of a simple Lie algebra g and the Inonu-Wigner contraction of p with respect to its decomposition into its standard Levi factor and its nilpotent radical : this is the Lie algebra which is isomorphic to p as a vector space, but where the nilpotent radical becomes an abelian ideal of this contraction. The study of the algebra of symmetric semi-invariants under the adjoint action associated with such a contraction was initiated in my paper entitled : Symmetric Semi-Invariants for some Inonu-Wigner Contractions-I, published in Transformation Groups, January 2025, wherein a lower bound for the formal character of this algebra was built, when the latter is well defined. Here in this paper we build an upper bound for this formal character, when p is a maximal parabolic subalgebra in a classical simple Lie algebra g in type B, when the Levi subalgebra of p is associated with the set of all simple roots without a simple root of even index with Bourbaki notation (we call this case the even case). We show that both bounds coincide. This provides a Weierstrass section for the algebra of symmetric semi-invariants associated with such a contraction and the polynomiality of such an algebra follows.

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arXiv2023

Symmetric Semi-invariants for some Inonu-Wigner contractions

Let $\mathfrak p$ be a proper parabolic subalgebra of a simple Lie algebra $\mathfrak g$. Writing $\mathfrak p=\mathfrak r\oplus \mathfrak m$, with $\mathfrak r$ being the Levi factor of $\mathfrak p$ and $\mathfrak m$ the nilpotent radical of $\mathfrak p$, we may consider the semi-direct product $\tilde\mathfrak p=\mathfrak r\ltimes(\mathfrak m)^a$ where $(\mathfrak m)^a$ is an abelian ideal of $\tilde\mathfrak p$, isomorphic to $\mathfrak m$ as an $\mathfrak r$-module. Then $\tilde\mathfrak p$ is a Lie algebra, which is a special case of Inönü-Wigner contraction and may be considered as a degeneration of the parabolic subalgebra $\mathfrak p$. Let $S(\tilde\mathfrak p)$ be the symmetric algebra of $\tilde\mathfrak p$ (it is equal to the symmetric algebra $S(\mathfrak p)$ of $\mathfrak p$) and consider the algebra of semi-invariants $Sy(\tilde\mathfrak p)\subset S(\tilde\mathfrak p)$ under the adjoint action of $\tilde\mathfrak p$. Using what we call a generalized PBW filtration on a highest weight irreducible representation $V(λ)$ of $\mathfrak g$, induced by the standard degree filtration on the enveloping algebra $U(\mathfrak m^-)$ of $\mathfrak m^-$, the nilpotent radical of the opposite parabolic subalgebra $\mathfrak p^-$ of $\mathfrak p$, one obtains a lower bound for the formel character of the algebra $Sy(\tilde\mathfrak p)$, when the latter is well defined.

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arXiv2020

Weierstrass sections for some truncated parabolic subalgebras

In this paper, using Bourbaki's convention, we consider a simple Lie algebra $\mathfrak g\subset\mathfrak g\mathfrak l_m$ of type B, C or D and a parabolic subalgebra $\mathfrak p$ of $\mathfrak g$ associated with a Levi factor composed essentially, on each side of the second diagonal, by successive blocks of size two, except possibly for the first and the last ones. Extending the notion of a Weierstrass section introduced by Popov to the coadjoint action of the truncated parabolic subalgebra associated with $\mathfrak p$, we construct explicitly Weierstrass sections, which give the polynomiality (when it was not yet known) for the algebra generated by semi-invariant polynomial functions on the dual space $\mathfrak p^*$ of $\mathfrak p$ and which allow to linearize the semi-invariant generators. Our Weierstrass sections require the construction of an adapted pair, which is the analogue of a principal $\mathfrak s\mathfrak l_2$-triple in the non reductive case.

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arXiv2017

Polynomiality for the Poisson centre of truncated maximal parabolic subalgebras

We show that the Poisson centre of truncated maximal parabolic subalgebras of a simple Lie algebra of type B, D and E_6 is a polynomial algebra. In roughly half of the cases the polynomiality of the Poisson centre was already known by a completely different method. For the rest of the cases, our approach is to construct an algebraic slice in the sense of Kostant given by an adapted pair and the computation of an improved upper bound for the Poisson centre.

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arXiv2015

Slices for maximal parabolic subalgebras of a semisimple Lie algebra

Let p be a maximal truncated parabolic subalgebra of a simple Lie Algebra. It was shown in many cases that the Poisson centre Y(p) is a polynomial algebra. We construct a slice for the coadjoint action of p, thus extending a theorem of Kostant. The role of the principal sl_2-triple is played by an adapted pair.

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arXiv2015

Adapted pairs in type $A$ and regular nilpotent elements

Let $\mathfrak g$ be a simple Lie algebra over an algebraically closed field $\bf k$ of characteristic zero and $\bf G$ its adjoint group. Let $\mathfrak q$ be a biparabolic subalgebra of $\mathfrak g$. The algebra $Sy(\mathfrak q)$ of semi-invariants on $\mathfrak q^*$ is polynomial in most cases, in particular when $\mathfrak g$ is simple of type $A$ or $C$. On the other hand $\mathfrak q$ admits a canonical truncation $\mathfrak q_Λ$ such that $Sy(\mathfrak q)=Sy(\mathfrak q_Λ)=Y(\mathfrak q_Λ)$ where $Y(\mathfrak q_Λ)$ denotes the algebra of invariant functions on $\mathfrak q_Λ^*$. An adapted pair for $\mathfrak q_Λ$ is a pair $(h,\,η)\in \mathfrak q_Λ\times\mathfrak q_Λ^*$ such that $η$ is regular and $(ad\,h)η=-η$. In a previous paper of A. Joseph (2008) adapted pairs for every truncated biparabolic subalgebra $\mathfrak q_Λ$ of a simple Lie algebra $\mathfrak g$ of type $A$ were constructed and then provide Weierstrass sections for $Y(\mathfrak q_Λ)$ in $\mathfrak q_Λ^*$. These latter are linear subvarieties $η+V$ of $\mathfrak q_Λ^*$ such that the restriction map induces an algebra isomorphism of $Y(\mathfrak q_Λ)$ onto the algebra of regular functions on $η+V$. Here we show that for each of the adapted pairs $(h,\,η)$ constructed in the paper mentioned above one can express $η$ as the image of a regular nilpotent element $y$ of $\mathfrak g^*$ under the restriction to $\mathfrak q$. Since $y$ must be a $\bf G$ translate of the standard regular nilpotent element defined in terms of the already chosen set $π$ of simple roots, one may attach to $y$ a unique element of the Weyl group. Ultimately one can then hope to be able to describe adapted pairs (in general) through the Weyl group.

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arXiv2015

Adapted pairs and Weierstrass sections

Adapted pairs and Weierstrass sections are central to the invariant theory associated to the action of an algebraic Lie algebra a on a finite dimensional vector space X. In this a need not be a semisimple Lie algebra. Here their general properties are described particularly when a is the canonical truncation of a biparabolic subalgebra of a simple Lie algebra and X is the dual of a.

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arXiv2010

Slices for biparabolics of index one

Let $\mathfrak a$ be an algebraic Lie subalgebra of a simple Lie algebra $\mathfrak g$ with index $\mathfrak a \leq \rank \mathfrak g$. Let $Y(\mathfrak a)$ denote the algebra of $\mathfrak a$ invariant polynomial functions on $\mathfrak a^*$. An algebraic slice for $\mathfrak a$ is an affine subspace $η+V$ with $η\in \mathfrak a^*$ and $V \subset \mathfrak a^*$ a subspace of dimension index $\mathfrak a$ such that restriction of function induces an isomorphism of $Y(\mathfrak a)$ onto the algebra $R[η+V]$ of regular functions on $η+V$. Slices have been obtained in a number of cases through the construction of an adapted pair $(h,η)$ in which $h \in\mathfrak a$ is ad-semisimple, $η$ is a regular element of $\mathfrak a^*$ which is an eigenvector for $h$ of eigenvalue minus one and $V$ is an $h$ stable complement to $(\ad \mathfrak a)η$ in $\mathfrak a^*$. The classical case is for $\mathfrak g$ semisimple. Yet rather recently many other cases have been provided. For example if $\mathfrak g$ is of type $A$ and $\mathfrak a$ is a "truncated biparabolic" or a centralizer. In some of these cases (particular when the biparabolic is a Borel subalgebra) it was found that $η$ could be taken to be the restriction of a regular nilpotent element in $\mathfrak g$. Moreover this calculation suggested how to construct slices outside type $A$ when no adapted pair exists. This article makes a first step in taking these ideas further. Specifically let $\mathfrak a$ be a truncated biparabolic of index one (and then $\mathfrak g$ is of type $A$). In this case it is shown that the second member of an adapted pair $(h,η)$ for $\mathfrak a$ is the restriction of a particularly carefully chosen regular nilpotent element of $\mathfrak g$.

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