Abstract

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).

Highlights

  • The ground field k is algebraically closed and char(k) = 0

  • In [PY], we presented compatible Poisson brackets related to a Z2-grading q = q0 ⊕ q1 and studied the respective subalgebra Z = Z(q, q0)

  • We proved that: — Z(g, g0) is a Poisson-commutative subalgebra of S(g)g0 having the maximal possible transcendence degree, which equals/2; — with only four exceptions related to exceptional Lie algebras, Z(g, g0) is a polynomial algebra whose algebraically independent generators are explicitly described; — if g is a classical Lie algebra and g0 contains a regular nilpotent element of g, Z(g, g0) is a maximal Poisson-commutative subalgebra of S(g)g0

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Summary

Introduction

In [PY], we presented compatible Poisson brackets related to a Z2-grading q = q0 ⊕ q1 and studied the respective subalgebra Z = Z(q, q0). Results of the present paper stem from a surprising observation that if q is equipped with a Zm-grading with any m 2, one can naturally construct a compatible Poisson bracket { , } (Section 2). In this case, all Poisson brackets in P are linear and there are two lines l1, l2 ⊂ k2 such that Ω = k2 \ (l1 ∪ l2) ⊂ Ωreg and the Lie algebras corresponding to (a, b) ∈ Ω are isomorphic to q. Our general reference for semisimple Lie groups and algebras is [Lie3]

Preliminaries on Poisson brackets and polynomial contractions
Contractions and invariants
Automorphisms of finite order and compatible Poisson brackets
Poisson-commutative subalgebras related to compatible Poisson brackets
Pencils of skew-symmetric matrices and differentials
On algebraically independent generators
On the maximality problem
The case of a cyclic permutation
Gaudin subalgebras
A Gaudin model consists of n copies of h and the Hamiltonians
Fixed-point subalgebras in the infinite-dimensional case
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