Non-trivial central extensions of Lie algebras of differential operators in two and higher dimensions

1991 ◽  
Vol 265 (1-2) ◽  
pp. 86-91 ◽  
Author(s):  
A.O. Radul
Keyword(s):  
Lie Algebras ◽  
Differential Operators ◽  
Higher Dimensions ◽  
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.

Spectral Methods

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Simple Model ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Linear Algebra ◽  
Spectral Methods ◽  
Differential Operators ◽  
Higher Dimensions ◽  
Standard Linear ◽  
Sommerfeld Equation ◽  
Spectral Discretization

Thenumerical solution of ordinary differential equations (ODEs)with boundary conditions is studied here. Functions are approximated by polynomials in a Chebychev basis. Sections then cover spectral discretization, sampling, interpolation, differentiation, integration, and the basic ODE. Following Trefethen et al., differential operators are approximated as rectangular matrices. Boundary conditions add additional rows that turn them into square matrices. These can then be diagonalized using standard linear algebra methods. After studying various simple model problems, this method is applied to the Orr–Sommerfeld equation, deriving results originally due to Orszag. The difficulties of pushing spectral methods to higher dimensions are outlined.

scholarly journals UNIVERSAL CALABI–YAU ALGEBRA: TOWARDS AN UNIFICATION OF COMPLEX GEOMETRY

2003 ◽  
Vol 18 (30) ◽  
pp. 5541-5612 ◽  
Author(s):  
F. ANSELMO ◽  
J. ELLIS ◽  
D. V. NANOPOULOS ◽  
G. VOLKOV
Keyword(s):  
Lie Algebras ◽  
Complex Geometry ◽  
Algebraic Approach ◽  
Higher Dimensions ◽  
Normal Algebra ◽  
Natural Extensions ◽  
Recurrence Formulae ◽  
Different Dimensions ◽  
Arbitrary Dimensions

We present a universal normal algebra suitable for constructing and classifying Calabi–Yau spaces in arbitrary dimensions. This algebraic approach includes natural extensions of reflexive weight vectors to higher dimensions, related to Batyrev's reflexive polyhedra, and their n-ary combinations. It also includes a "dual" construction based on the Diophantine decomposition of invariant monomials, which provides explicit recurrence formulae for the numbers of Calabi–Yau spaces in arbitrary dimensions with Weierstrass, K3, etc., fibrations. Our approach also yields simple algebraic relations between chains of Calabi–Yau spaces in different dimensions, and concrete visualizations of their singularities related to Cartan–Lie algebras. This Universal Calabi–Yau algebra is a powerful tool for deciphering the Calabi–Yau genome in all dimensions.

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.

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