An operadic approach to deformation quantization of compatible Poisson brackets, I

2007 ◽  
Vol 01 (02) ◽  
Author(s):  
Vladimir DOTSENKO
Keyword(s):  
Deformation Quantization ◽  
Poisson Brackets ◽  
Compatible Poisson Brackets

scholarly journals PERIODIC AUTOMORPHISMS, COMPATIBLE POISSON BRACKETS, AND GAUDIN SUBALGEBRAS

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).

Classical r-matrices and compatible Poisson brackets for coupled KdV systems

1989 ◽  
Vol 17 (1) ◽  
pp. 25-29 ◽  
Author(s):  
A. P. Fordy ◽  
A. G. Reyman ◽  
M. A. Semenov-Tian-Shansky
Keyword(s):  
Poisson Brackets ◽  
Compatible Poisson Brackets

scholarly journals DEFORMATION QUANTIZATION OF CERTAIN NONLINEAR POISSON STRUCTURES

1998 ◽  
Vol 09 (05) ◽  
pp. 599-621 ◽  
Author(s):  
BYUNG-JAY KAHNG
Keyword(s):  
Poisson Bracket ◽  
Lie Algebra ◽  
Deformation Quantization ◽  
Dual Space ◽  
Poisson Brackets ◽  
Poisson Structures ◽  
Central Extensions ◽  
C Algebras ◽  
Lie Bracket ◽  
Compact Quantum Groups

As a generalization of the linear Poisson bracket on the dual space of a Lie algebra, we introduce certain nonlinear Poisson brackets which are "cocycle perturbations" of the linear Poisson bracket. We show that these special Poisson brackets are equivalent to Poisson brackets of central extension type, which resemble the central extensions of an ordinary Lie bracket via Lie algebra cocycles. We are able to formulate (strict) deformation quantizations of these Poisson brackets by means of twisted group C*-algebras. We also indicate that these deformation quantizations can be used to construct some specific non-compact quantum groups.

BUNDLES OF LIE ALGEBRAS AND COMPATIBLE POISSON BRACKETS

2002 ◽  
Vol 17 (06n07) ◽  
pp. 946-950
Author(s):  
A. B. YANOVSKI
Keyword(s):  
Lie Algebras ◽  
Poisson Brackets ◽  
Algebraic Structures ◽  
Compatible Poisson Brackets ◽  
Lie Brackets

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.

Sign in / Sign up

Export Citation Format

Share Document