Lie algebras of differential operators, their central extensions, and W-algebras

1991 ◽  
Vol 25 (1) ◽  
pp. 25-39 ◽  
Author(s):  
A. O. Radul
Keyword(s):  
Lie Algebras ◽  
Differential Operators ◽  
Central Extensions

scholarly journals Formal Pseudodifferential Operators in One and Several Variables, Central Extensions, and Integrable Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-16 ◽  
Author(s):  
Jarnishs Beltran ◽  
Enrique G. Reyes
Keyword(s):  
Lie Algebras ◽  
Nonlinear Equations ◽  
Integrable Systems ◽  
Differential Operators ◽  
Pseudodifferential Operators ◽  
Central Extensions ◽  
Several Variables ◽  
Zero Curvature ◽  
Manin Triples

We review some aspects of the theory of Lie algebras of (twisted and untwisted) formal pseudodifferential operators in one and several variables in a general algebraic context. We focus mainly on the construction and classification of nontrivial central extensions. As applications, we construct hierarchies of centrally extended Lie algebras of formal differential operators in one and several variables, Manin triples and hierarchies of nonlinear equations in Lax and zero curvature form.

Constructing lie algebras of first-order differential operators

1997 ◽  
Vol 40 (6) ◽  
pp. 525-530 ◽  
Author(s):  
I. V. Shirokov
Keyword(s):  
Lie Algebras ◽  
Differential Operators ◽  
First Order

scholarly journals Integrating central extensions of Lie algebras via Lie 2-groups

2016 ◽  
Vol 18 (6) ◽  
pp. 1273-1320 ◽  
Author(s):  
Christoph Wockel ◽  
Chenchang Zhu
Keyword(s):  
Lie Algebras ◽  
Central Extensions

scholarly journals KONTSEVICH'S UNIVERSAL FORMULA FOR DEFORMATION QUANTIZATION AND THE CAMPBELL–BAKER–HAUSDORFF FORMULA

2000 ◽  
Vol 11 (04) ◽  
pp. 523-551 ◽  
Author(s):  
VINAY KATHOTIA
Keyword(s):  
Lie Algebras ◽  
Deformation Quantization ◽  
Differential Operators ◽  
Main Tool ◽  
Graphical Analysis ◽  
Poisson Structures ◽  
Nilpotent Lie Algebras ◽  
Enveloping Algebra ◽  
Universal Formula ◽  
The Given

We relate a universal formula for the deformation quantization of Poisson structures (⋆-products) on ℝd proposed by Maxim Kontsevich to the Campbell–Baker–Hausdorff (CBH) formula. We show that Kontsevich's formula can be viewed as exp (P) where P is a bi-differential operator that is a deformation of the given Poisson structure. For linear Poisson structures (duals of Lie algebras) his product takes the form exp (C+L) where exp (C) is the ⋆-product given by the universal enveloping algebra via symmetrization, essentially the CBH formula. This is established by showing that the two products are identical on duals of nilpotent Lie algebras where the operator L vanishes. This completely determines part of Kontsevich's formula and leads to a new scheme for computing the CBH formula. The main tool is a graphical analysis for bi-differential operators and the computation of certain iterated integrals that yield the Bernoulli numbers.

scholarly journals Central extensions of the quasi-orthogonal Lie algebras

1998 ◽  
Vol 31 (5) ◽  
pp. 1373-1394 ◽  
Author(s):  
J A de Azcárraga ◽  
F J Herranz ◽  
J C Pérez Bueno ◽  
M Santander
Keyword(s):  
Lie Algebras ◽  
Central Extensions

Pseudo-Differential Operators $(\psi{\cal D}{\cal O})$ Algebra and Certain Infinite-Dimensional Lie Algebras

1997 ◽  
Vol 12 (22) ◽  
pp. 1589-1595 ◽  
Author(s):  
E. H. El Kinani
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Differential Operators ◽  
Quantum Plane ◽  
Infinite Dimensional ◽  
Pseudo Differential Operators ◽  
Physics Literature

The class of pseudo-differential operators Lie algebra [Formula: see text] on the quantum plane [Formula: see text] is introduced. The embedding of certain infinite-dimensional Lie algebras which occur in the physics literature in [Formula: see text] is discussed as well as the correspondence between [Formula: see text] and [Formula: see text] as k→+∞ is examined.

Central extensions of Lie algebras graded by finite root systems

2000 ◽  
Vol 316 (3) ◽  
pp. 499-527 ◽  
Author(s):  
Bruce Allison ◽  
Georgia Benkart ◽  
Yun Gao
Keyword(s):  
Lie Algebras ◽  
Root Systems ◽  
Central Extensions
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