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Differential Invariants in Algebra Read chapter Read first chapter

2023 | OriginalPaper | Chapter

Lectures on Poisson Algebras

Authors : Vladimir Rubtsov, Radek Suchánek

Published in: Groups, Invariants, Integrals, and Mathematical Physics

Publisher: Springer Nature Switzerland

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Abstract

The notion of a Poisson algebra was probably introduced in the first time by A.M. Vinogradov and J. S. Krasil’shchik in 1975 under the name “canonical algebra” and by J. Braconnier in his short note “Algèbres de Poisson” (Comptes rendus Ac.Sci) in 1977.

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previous chapter Differential Invariants in Algebra
next chapter Some Remarks on Multisymplectic and Variational Nature of Monge-Ampère Equations in Dimension Four
Footnotes
1
Given an increasing sequence of ideals in \(\mathcal {A}\), \(\mathcal {I}_1 \subseteq \mathcal {I}_2 \subseteq \ldots \), there always exists \(k \in \mathbb {N}\) such that \(\mathcal {I}_k = \mathcal {I}_{k+n} \) for all \(n \in \mathbb {N}\).
 
2
s1, s2 ∈ S ⇒ s1s2 ∈ S.
 
3
In this notation, ka can be interpreted equivalently as action of \(\mathbb {K}\) on \(\mathcal {A}\), or as a multiplication in \(\mathcal {A}\) after identifying \(\mathbb {K}\) with \(\iota (\mathbb {K})\). Of course we have k1 = 1k = k.
 
4
Since for a ≠ 0 : ka = 0 ⇒ k = 0.
 
5
That is, \(\mathcal {A}\) acts on \(\mathcal {M}\) from left and right.
 
6
Notice that using the action of \(\mathfrak {g}\) on itself given by multiplication from the left, we can identify elements in \(\mathfrak {g}\) as endomorphisms of \(\mathfrak {g}\), h↦lh. Then we have δg(lh)(u) = h(gu) − gh(u) = (adgh)u, thus \(\delta _g|{ }_{\mathfrak {g}} \equiv \operatorname {ad}_g\).
 
7
Every element of \(\mathfrak {g}\) serves as a variable. One can think of \(\mathbb {K} [\mathfrak {g}]\) as \(\mathbb {K}[g_1, g_2, \ldots ]\) for all \(g_i \in \mathfrak {g}\).
 
8
The kernel of the representation, as a homomorphism from \(\mathfrak {g}\), is trivial. Example: \(\mathfrak {g}\) acts faithfully on \(C^{\infty }(\mathfrak {g})\). Another example: \(\mathfrak {g}\) is a matrix algebra (n × n matrices), then \(\mathfrak {g}\) acts faithfully on \(\mathbb {R}^n\).
 
9
This is the evaluation map \(F_X \colon \mathfrak {g}^* \to \mathbb {R}\) (defined above), FX(ξ) = ξ(X).
 
10
Meaning \(s \in \mathfrak {X}^{|s|} (M)\).
 
11
See Definition 2.13 for the definition of Poisson center \(\operatorname {Z}_{P}(\mathcal {A})\).
 
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Metadata
Title
Lectures on Poisson Algebras
Authors
Vladimir Rubtsov
Radek Suchánek
Copyright Year
2023
Book
Groups, Invariants, Integrals, and Mathematical Physics

Print ISBN: 978-3-031-25665-3

Electronic ISBN: 978-3-031-25666-0

Copyright Year: 2023

DOI
https://doi.org/10.1007/978-3-031-25666-0_2

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