[Paper Review] Adapted pairs and Weierstrass sections

Florence Fauquant-Millet, Anthony Joseph|arXiv (Cornell University)|Mar 9, 2015

Advanced Algebra and Geometry17 references3 citations

TL;DR

This paper investigates adapted pairs and Weierstrass sections in the context of invariant theory for algebraic Lie algebras, particularly focusing on canonical truncations of biparabolic subalgebras of simple Lie algebras. It establishes conditions under which Weierstrass sections exist and proves that the algebra of invariants $ Y(rak{a}) $ is polynomial, extending Kostant's linearisation result beyond semisimple Lie algebras.

ABSTRACT

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.

Motivation & Objective

• To generalize Kostant's linearisation of invariants to non-semisimple Lie algebras, particularly truncated biparabolic subalgebras.
• To establish conditions under which Weierstrass sections exist for the coadjoint action of an algebraic Lie algebra $\frak{a}$ on $\frak{a}^*$.
• To investigate the polynomiality of the invariant algebra $ Y(\frak{a}) = S(\frak{a})^\bf{A} $ when $\frak{a}$ is not semisimple.
• To clarify the role of adapted pairs and the codimension of singular loci in determining whether a linear slice is a Weierstrass section.
• To provide a framework for understanding invariant generators in cases where no $\mathfrak{sl}_2$ triple exists, such as in type $A$ and $C$ truncated biparabolics.

Proposed method

• Utilizes the concept of adapted pairs $ (e,h) $, where $ e $ is a nilpotent element and $ h $ is a semisimple element satisfying $ [h,e] = -2e $, to construct linear slices.
• Applies the notion of Weierstrass sections as linear subvarieties $ \eta + V \subset \frak{a}^* $ such that the restriction map induces an isomorphism between $ Y(\frak{a}) $ and the regular functions on $ \eta + V $.
• Analyzes the coadjoint action of the adjoint group $ \bf{A} $ on $ \frak{a}^* $, focusing on orbit transversality and the structure of regular and singular loci.
• Employs the Gelfand-Kirillov dimension of $ Y(\frak{a}) $, equated to the index $ \ell(\frak{a}) $, to assess the growth rate and support polynomiality.
• Uses explicit computations in the filiform Lie algebra of dimension 5 to demonstrate that a linear slice may fail to be a Weierstrass section if orbits do not intersect transversally or if the singular locus has codimension 1.
• Applies results from Popov and Chevalley-Rosenlicht to relate the existence of Weierstrass sections to the smoothness of the restriction map and the absence of proper semi-invariants.

Experimental results

Research questions

• RQ1 Under what conditions does a linear slice $ \eta + V \subset \frak{a}^* $ become a Weierstrass section for the coadjoint action of $ \bf{A} $?
• RQ2 Can the algebra of invariants $ Y(\frak{a}) $ be shown to be polynomial for non-semisimple Lie algebras such as truncated biparabolic subalgebras?
• RQ3 How does the failure of transversality or the codimension of the singular locus affect the existence of a Weierstrass section?
• RQ4 To what extent can Kostant's linearisation of invariants be generalized to algebras lacking a principal $\mathfrak{sl}_2 $ triple?
• RQ5 What is the role of adapted pairs in constructing Weierstrass sections when $\frak{a}$ is not semisimple?

Key findings

• For canonical truncations of biparabolic subalgebras in type $A$ and $C$, the algebra of invariants $ Y(\frak{a}) $ is polynomial, extending classical results beyond semisimple Lie algebras.
• A Weierstrass section exists if and only if the restriction of $ S(\frak{a}^*) $ to $ \eta + V $ induces an isomorphism with $ Y(\frak{a}) $, which holds precisely when the slice intersects all regular orbits transversally and the singular locus has codimension at least 2.
• In the example of the 5-dimensional filiform Lie algebra, $ Y(\frak{a}) $ is not polynomial, and the linear slice $ V \setminus D $ fails to be a Weierstrass section due to non-transverse orbit intersections and codimension-1 singularities.
• The condition that $ \bf{A}.(\eta + V) \setminus (\eta + V)_{\text{reg}} $ has codimension $ \geq 2 $ is necessary for $ \eta + V $ to be a Weierstrass section, and this fails when the singular locus is of codimension 1.
• The fundamental semi-invariant of the filiform algebra is scalar, implying that $ \frak{a}^* \setminus \frak{a}^*_{\text{reg}} $ has codimension $ \geq 2 $, yet $ \mathscr{S} $ does not meet all regular orbits, so it is not a Weierstrass section.
• The paper shows that the existence of a Weierstrass section implies polynomiality of $ Y(\frak{a}) $, but the converse does not hold without additional geometric conditions on the slice and orbit structure.