On the central extensions of Poisson brackets of hydrodynamic type

2003 ◽  
Vol 36 (9) ◽  
pp. 2261-2269
Author(s):  
Chengming Bai ◽  
Daoji Meng ◽  
Hongbiao Zhang
Keyword(s):  
Hydrodynamic Type ◽  
Poisson Brackets ◽  
Central Extensions

scholarly journals SECOND-ORDER DEFORMATIONS OF HYDRODYNAMIC-TYPE POISSON BRACKETS

2009 ◽  
Vol 51 (A) ◽  
pp. 75-82
Author(s):  
JAMES T. FERGUSON
Keyword(s):  
Differential Operator ◽  
Poisson Bracket ◽  
Hamiltonian Structure ◽  
Second Order ◽  
Hydrodynamic Type ◽  
Poisson Brackets ◽  
Geometric Type

AbstractThis paper is concerned with the properties of differential-geometric-type Poisson brackets specified by a differential operator of degree 2. It also considers the conditions required for such a Poisson bracket to form a bi-Hamiltonian structure with a hydrodynamic-type Poisson bracket.

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.

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