Poisson Commutative Subalgebra Research Articles

Automorphisms of finite order, periodic contractions, and Poisson-commutative subalgebras of {mathcal {S}}({mathfrak g})

Let {mathfrak g} be a semisimple Lie algebra, vartheta in textsf{Aut}({mathfrak g}) a finite order automorphism, and {mathfrak g}_0 the subalgebra of fixed points of vartheta . Recently, we noticed that using vartheta one can construct a pencil of compatible Poisson brackets on {mathcal {S}}({mathfrak g}), and thereby a ‘large’ Poisson-commutative subalgebra {mathcal {Z}}({mathfrak g},vartheta ) of {mathcal {S}}({mathfrak g})^{{mathfrak g}_0}. In this article, we study invariant-theoretic properties of ({mathfrak g},vartheta ) that ensure good properties of {mathcal {Z}}({mathfrak g},vartheta ). Associated with vartheta one has a natural Lie algebra contraction {mathfrak g}_{(0)} of {mathfrak g} and the notion of a good generating system (=g.g.s.) in {mathcal {S}}({mathfrak g})^{mathfrak g}. We prove that in many cases the equality mathrm{ind,}{mathfrak g}_{(0)}=mathrm{ind,}{mathfrak g} holds and {mathcal {S}}({mathfrak g})^{mathfrak g} has a g.g.s. According to V. G. Kac’s classification of finite order automorphisms (1969), vartheta can be represented by a Kac diagram, mathcal {K}(vartheta ), and our results often use this presentation. The most surprising observation is that {mathfrak g}_{(0)} depends only on the set of nodes in mathcal {K}(vartheta ) with nonzero labels, and that if vartheta is inner and a certain label is nonzero, then {mathfrak g}_{(0)} is isomorphic to a parabolic contraction of {mathfrak g}.

PERIODIC AUTOMORPHISMS, COMPATIBLE POISSON BRACKETS, AND GAUDIN SUBALGEBRAS

Let \U0001d52e be a finite-dimensional Lie algebra. The symmetric algebra (\U0001d52e) is equipped with the standard Lie–Poisson bracket. In this paper, we elaborate on a surprising observation that one naturally associates the second compatible Poisson bracket on (\U0001d52e) to any finite order automorphism ϑ of \U0001d52e. We study related Poisson-commutative subalgebras (\U0001d52e; ϑ) of \U0001d4ae(\U0001d52e) and associated Lie algebra contractions of \U0001d52e. To obtain substantial results, we have to assume that \U0001d52e = \U0001d524 is semisimple. Then we can use Vinberg’s theory of ϑ-groups and the machinery of Invariant Theory.If \U0001d524 = \U0001d525⊕⋯⊕\U0001d525 (sum of k copies), where \U0001d525 is simple, and ϑ is the cyclic permutation, then we prove that the corresponding Poisson-commutative subalgebra (\U0001d52e; ϑ) is polynomial and maximal. Furthermore, we quantise this (\U0001d524; ϑ) using a Gaudin subalgebra in the enveloping algebra \U0001d4b0(\U0001d524).

Compatible Poisson brackets associated with 2‐splittings and Poisson commutative subalgebras of S(g)

Let ${\mathcal S}(\mathfrak g)$ be the symmetric algebra of a reductive Lie algebra $\mathfrak g$ equipped with the standard Poisson structure. If ${\mathcal C}\subset\mathcal S(\mathfrak g)$ is a Poisson-commutative subalgebra, then ${\rm trdeg\,}{\mathcal C}\le\boldsymbol{b}(\mathfrak g)$, where $\boldsymbol{b}(\mathfrak g)=(\dim\mathfrak g+{\rm rk}\mathfrak g)/2$. We present a method for constructing the Poisson-commutative subalgebra $\mathcal Z_{\langle\mathfrak h,\mathfrak r\rangle}$ of transcendence degree $\boldsymbol{b}(\mathfrak g)$ via a vector space decomposition $\mathfrak g=\mathfrak h\oplus\mathfrak r$ into a sum of two spherical subalgebras. There are some natural examples, where the algebra $\mathcal Z_{\langle\mathfrak h,\mathfrak r\rangle}$ appears to be polynomial. The most interesting case is related to the pair $(\mathfrak b,\mathfrak u_-)$, where $\mathfrak b$ is a Borel subalgebra of $\mathfrak g$. Here we prove that ${\mathcal Z}_{\langle\mathbb b,\mathbb u_-\rangle}$ is maximal Poisson-commutative and is complete on every regular coadjoint orbit in $\mathfrak g^*$. Other series of examples are related to decompositions associated with involutions of $\mathfrak g$.

Poisson-Commutative Subalgebras of 𝒮(𝔤) associated with Involutions

Abstract The symmetric algebra ${\mathcal{S}}({{\mathfrak{g}}})$ of a reductive Lie algebra ${{\mathfrak{g}}}$ is equipped with the standard Poisson structure, that is, the Lie–Poisson bracket. Poisson-commutative subalgebras of ${\mathcal{S}}({{\mathfrak{g}}})$ attract a great deal of attention because of their relationship to integrable systems and, more recently, to geometric representation theory. The transcendence degree of a Poisson-commutative subalgebra ${\mathcal C}\subset{\mathcal{S}}({{\mathfrak{g}}})$ is bounded by the “magic number” ${\boldsymbol{b}}({{\mathfrak{g}}})$ of ${{\mathfrak{g}}}$. There are two classical constructions of $\mathcal C$ with ${\textrm{tr.deg}}\,{\mathcal C}={\boldsymbol{b}}({{\mathfrak{g}}})$. The 1st one is applicable to $\mathfrak{gl}_n$ and $\mathfrak{so}_n$ and uses the Gelfand–Tsetlin chains of subalgebras. The 2nd one is known as the “argument shift method” of Mishchenko–Fomenko. We generalise the Gelfand–Tsetlin approach to chains of almost arbitrary symmetric subalgebras. Our method works for all types. Starting from a symmetric decompositions ${{\mathfrak{g}}}={{\mathfrak{g}}}_0\oplus{{\mathfrak{g}}}_1$, Poisson-commutative subalgebras ${{\mathcal{Z}}},\tilde{{\mathcal{Z}}}\subset{\mathcal{S}}({{\mathfrak{g}}})^{{{\mathfrak{g}}}_0}$ of the maximal possible transcendence degree are constructed. If the ${{\mathbb{Z}}}_2$-contraction ${{\mathfrak{g}}}_0\ltimes{{\mathfrak{g}}}_1^{\textsf{ab}}$ has a polynomial ring of symmetric invariants, then $\tilde{{\mathcal{Z}}}$ is a polynomial maximal Poisson-commutative subalgebra of ${\mathcal{S}}({{\mathfrak{g}}})^{{{\mathfrak{g}}}_0}$ and its free generators are explicitly described.

On the Fibres of Mishchenko-Fomenko Systems

This work is concerned with Mishchenko and Fomenko's celebrated theory of completely integrable systems on a complex semisimple Lie algebra $\mathfrak{g}$. Their theory associates a maximal Poisson-commutative subalgebra of $\mathbb{C}[\mathfrak{g}]$ to each regular element $a\in\mathfrak{g}$, and one can assemble free generators of this subalgebra into a moment map $F_a:\mathfrak{g}\rightarrow\mathbb{C}^b$. We examine the structure of fibres in Mishchenko--Fomenko systems, building on the foundation laid by Bolsinov, Charbonnel--Moreau, Moreau, and others. This includes proving that the critical values of $F_a$ have codimension $1$ or $2$ in $\mathbb{C}^b$, and that each codimension is achievable in examples. Our results on singularities make use of a subalgebra $\mathfrak{b}^a\subseteq\mathfrak{g}$, defined to be the intersection of all Borel subalgebras of $\mathfrak{g}$ containing $a$. In the case of a non-nilpotent $a\in\mathfrak{g}_{\text{reg}}$ and an element $x\in\mathfrak{b}^a$, we prove the following: $x+[\mathfrak{b}^a,\mathfrak{b}^a]$ lies in the singular locus of $F_a^{-1}(F_a(x))$, and the fibres through points in $\mathfrak{b}^a$ form a $\mathrm{rank}(\mathfrak{g})$-dimensional family of singular fibres. We next consider the irreducible components of our fibres, giving a systematic way to construct many components via Mishchenko--Fomenko systems on Levi subalgebras $\mathfrak{l}\subseteq\mathfrak{g}$. In addition, we obtain concrete results on irreducible components that do not arise from the aforementioned construction. Our final main result is a recursive formula for the number of irreducible components in $F_a^{-1}(0)$, and it generalizes a result of Charbonnel--Moreau. Illustrative examples are included at the end of this paper.

Poisson-commutative subalgebras and complete integrability on non-regular coadjoint orbits and flag varieties

The purpose of this paper is to bring together various loose ends in the theory of integrable systems. For a semisimple Lie algebra \({\mathfrak {g}}\), we obtain several results on the completeness of homogeneous Poisson-commutative subalgebras of \({\mathscr {S}}({\mathfrak {g}})\) on coadjoint orbits. This concerns, in particular, Gelfand–Tsetlin and Mishchenko–Fomenko subalgebras. Our results reveal the crucial role of nilpotent orbits and sheets in \({\mathfrak {g}}\simeq {\mathfrak {g}}^{*}\).

The Maximal Abelian Dimension of a Lie Algebra, Rentschler’s Property and Milovanov’s Conjecture

A finite dimensional Lie algebra L with magic number c(L) is said to satisfy Rentschler’s property if it admits an abelian Lie subalgebra H of dimension at least c(L) − 1. We study the occurrence of this new property in various Lie algebras, such as nonsolvable, solvable, nilpotent, metabelian and filiform Lie algebras. Under some mild condition H gives rise to a complete Poisson commutative subalgebra of the symmetric algebra S(L). Using this, we show that Milovanov’s conjecture holds for the filiform Lie algebras of type Ln, Qn, Rn, Wn and also for all filiform Lie algebras of dimension at most eight. For the latter the Poisson center of these Lie algebras is determined.

Deformation quantization of integrable systems

In this paper we address the following question: is it always possible to choose a deformation quantization of a Poisson algebra A so that certain Poisson-commutative subalgebra C in it remains commutative? We define a series of cohomological obstructions to this, that take values in the Hochschild cohomology of C with coefficients in A. In some particular case of the pair (A,C) we reduce these classes to the classes of the Poisson relative cohomology of the Hochschild cohomology. We show, that in the case, when the algebra C is polynomial, these obstructions coincide with the previously known ones, those which were defined by Garay and van Straten.

The Poisson center and polynomial, maximal Poisson commutative subalgebras, especially for nilpotent Lie algebras of dimension at most seven

Let g be a finite dimensional Lie algebra over an algebraically closed field k of characteristic zero. We collect some general results on the Poisson center of S(g), including some simple criteria regarding its polynomiality, and also on certain Poisson commutative subalgebras of S(g). These facts are then used to complete our previous work on the subject (Ooms, 2009 [O5, 5]), i.e. to give an explicit description for the Poisson center of all indecomposable, nilpotent Lie algebras of dimension at most seven. Among other things, we also provide a polynomial, maximal Poisson commutative subalgebra of S(g), enjoying additional properties. As a by-product we show that a conjecture by Milovanov is valid in this situation. These results easily carry over to the enveloping algebra U(g).

Rigid subsets of symplectic manifolds

AbstractWe show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the others. We also find new examples of rigidity of intersections involving, in particular, specific fibers of moment maps of Hamiltonian torus actions, monotone Lagrangian submanifolds (following the works of P. Albers and P. Biran-O. Cornea) as well as certain, possibly singular, sets defined in terms of Poisson-commutative subalgebras of smooth functions. In addition, we get some geometric obstructions to semi-simplicity of the quantum homology of symplectic manifolds. The proofs are based on the Floer-theoretical machinery of partial symplectic quasi-states.

Uniqueness of higher Gaudin Hamiltonians

For any semisimple Lie algebra g , the universal enveloping algebra of the infinite-dimensional pro-nilpotent Lie algebra g _:= g ⊗ t − 1 ℂ [ t − 1 ] contains a large commutative subalgebra A ⊂ U ( g _ ) . This subalgebra comes from the center of the universal enveloping of the affine Kac-Moody algebra g ^ at the critical level by the AKS-scheme. In this note we show that the corresponding “classical” Poisson-commutative subalgebra gr A ⊂ S ( g _ ) is the Poisson centralizer of its simplest quadratic element, and deduce from this that the “quantum” subalgebra A ⊂ U ( g _ ) is uniquely determined by the classical one. As an application, we show that Feigin-Frenkel-Reshetikhin's and Talalaev-Chervov's constructions of higher Hamiltonians of the Gaudin model give the same family of commuting operators.

Maximal Poisson Commutative subalgebras for truncated parabolic subalgebras of maximal index in sln

We show that several properties of the semisimple algebras carry over to a certain family of parabolic subalgebras of maximal index in sln. More precisely we prove an analogue of Kostant's slice theorem [B. Kostant, Amer. J. Math. 85 (1963), 327-404] for these algebras and construct a maximal Poisson commutative subalgebra in the symmetric algebra, following the theory presented in [A.S. Mishchenko and A.T. Fomenko, Math. USSR-Izv. 12 (1978), 371-389]. These results are quite remarkable since these algebras do not admit appropriate sl2-triples.

The argument shift method and maximal commutative subalgebras of Poisson algebras

Let $S$ be the symmetric algebra of an algebraic Lie algebra. We provide a sufficient condition for the maximality of Poisson commutative subalgebras of $S$ obtained by the argument shift method.