COMPATIBLE POISSON BRACKETS ON LIE ALGEBRAS AND COMPLETENESS OF FAMILIES OF FUNCTIONS IN INVOLUTION

We consider the mechanism of obtaining compatible Poisson-Lie tensors, based on the existence of some special algebraic structures - families of Lie brackets defined on one and the same space.

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PERIODIC AUTOMORPHISMS, COMPATIBLE POISSON BRACKETS, AND GAUDIN SUBALGEBRAS

Transformation Groups ◽

10.1007/s00031-021-09650-3 ◽

2021 ◽

Author(s):

DMITRI I. PANYUSHEV ◽

OKSANA S. YAKIMOVA

Keyword(s):

Poisson Bracket ◽

Lie Algebra ◽

Invariant Theory ◽

Cyclic Permutation ◽

Poisson Brackets ◽

Enveloping Algebra ◽

Order Automorphism ◽

Finite Dimensional ◽

Compatible Poisson Brackets ◽

Commutative Subalgebras

AbstractLet 𝔮 be a finite-dimensional Lie algebra. The symmetric algebra (𝔮) 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 (𝔮) to any finite order automorphism ϑ of 𝔮. We study related Poisson-commutative subalgebras (𝔮; ϑ) of 𝒮(𝔮) and associated Lie algebra contractions of 𝔮. To obtain substantial results, we have to assume that 𝔮 = 𝔤 is semisimple. Then we can use Vinberg’s theory of ϑ-groups and the machinery of Invariant Theory.If 𝔤 = 𝔥⊕⋯⊕𝔥 (sum of k copies), where 𝔥 is simple, and ϑ is the cyclic permutation, then we prove that the corresponding Poisson-commutative subalgebra (𝔮; ϑ) is polynomial and maximal. Furthermore, we quantise this (𝔤; ϑ) using a Gaudin subalgebra in the enveloping algebra 𝒰(𝔤).

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A note on the algebra of Poisson brackets

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100061922 ◽

1984 ◽

Vol 96 (1) ◽

pp. 45-60 ◽

Cited By ~ 2

Author(s):

C. J. Atkin

Keyword(s):

Poisson Bracket ◽

Lie Algebra ◽

Lie Algebras ◽

Symplectic Manifold ◽

Vector Fields ◽

Poisson Brackets ◽

Infinite Dimensional

In a long sequence of notes in the Comptes Rendus and elsewhere, and in the papers [1], [2], [3], [6], [7], Lichnerowicz and his collaborators have studied the ‘classical infinite-dimensional Lie algebras’, their derivations, automorphisms, co-homology, and other properties. The most familiar of these algebras is the Lie algebra of C∞ vector fields on a C∞ manifold. Another is the Lie algebra of ‘Poisson brackets’, that is, of C∞ functions on a C∞ symplectic manifold, with the Poisson bracket as composition; some questions concerning this algebra are of considerable interest in the theory of quantization – see, for instance, [2] and [3].

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