Poisson Brackets and Structure of Nongraded Hamiltonian Lie Algebras Related to Locally-Finite Derivations

AbstractXu introduced a class of nongraded Hamiltonian Lie algebras. These Lie algebras have a Poisson bracket structure. In this paper, the isomorphism classes of these Lie algebras are determined by employing a “sandwich” method and by studying some features of these Lie algebras. It is obtained that two Hamiltonian Lie algebras are isomorphic if and only if their corresponding Poisson algebras are isomorphic. Furthermore, the derivation algebras and the second cohomology groups are determined.

Download Full-text

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].

Download Full-text

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).

Download Full-text

A Property of Locally Finite Lie Algebras

Journal of the London Mathematical Society ◽

10.1112/jlms/s2-3.2.334 ◽

1971 ◽

Vol s2-3 (2) ◽

pp. 334-340 ◽

Cited By ~ 3

Author(s):

I. N. Stewart

Keyword(s):

Lie Algebras ◽

Locally Finite

Download Full-text

Nambu–Poisson bracket on superspace

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887818501906 ◽

2018 ◽

Vol 15 (11) ◽

pp. 1850190 ◽

Cited By ~ 2

Author(s):

Viktor Abramov

Keyword(s):

Poisson Bracket ◽

Jacobi Identity ◽

Leibniz Rule ◽

Smooth Functions ◽

Poisson Algebras ◽

Fundamental Identity ◽

Odd Degree ◽

New Formula ◽

Usual Formula

We propose an extension of [Formula: see text]-ary Nambu–Poisson bracket to superspace [Formula: see text] and construct by means of superdeterminant a family of Nambu–Poisson algebras of even degree functions, where the parameter of this family is an invertible transformation of Grassmann coordinates in superspace [Formula: see text]. We prove in the case of the superspaces [Formula: see text] and [Formula: see text] that our [Formula: see text]-ary bracket, defined with the help of superdeterminant, satisfies the conditions for [Formula: see text]-ary Nambu–Poisson bracket, i.e. it is totally skew-symmetric and it satisfies the Leibniz rule and the Filippov–Jacobi identity (fundamental identity). We study the structure of [Formula: see text]-ary bracket defined with the help of superdeterminant in the case of superspace [Formula: see text] and show that it is the sum of usual [Formula: see text]-ary Nambu–Poisson bracket and a new [Formula: see text]-ary bracket, which we call [Formula: see text]-bracket, where [Formula: see text] is the product of two odd degree smooth functions.

Download Full-text