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.

scholarly journals Classification of nilpotent associative algebras of small dimension

2018 ◽  
Vol 28 (01) ◽  
pp. 133-161 ◽  
Author(s):  
Willem A. De Graaf
Keyword(s):  
Lie Algebras ◽  
Small Dimension ◽  
Associative Algebras ◽  
Central Extensions ◽  
Nilpotent Lie Algebras

We classify nilpotent associative algebras of dimensions up to [Formula: see text] over any field. This is done by constructing the nilpotent associative algebras as central extensions of algebras of smaller dimension, analogous to methods known for nilpotent Lie algebras.

scholarly journals CLASSIFICATION OF SIX-DIMENSIONAL REAL DRINFELD DOUBLES

2002 ◽  
Vol 17 (28) ◽  
pp. 4043-4067 ◽  
Author(s):  
LIBOR ŠNOBL ◽  
LADISLAV HLAVATÝ
Keyword(s):  
Lie Algebras ◽  
Explicit Form ◽  
Complete List ◽  
Invariant Form ◽  
Manin Triples

Starting from the classification of real Manin triples we look for those that are isomorphic as six-dimensional Drinfeld doubles i.e. Lie algebras with the ad-invariant form used for construction of the Manin triples. We use several invariants of the Lie algebras to distinguish the nonisomorphic structures and give the explicit form of maps between Manin triples that are decompositions of isomorphic Drinfeld doubles. The result is a complete list of six-dimensional real Drinfeld doubles. It consists of 22 classes of nonisomorphic Drinfeld doubles.

An algebraic model of graded calculus of variations

1987 ◽  
Vol 101 (1) ◽  
pp. 151-166 ◽  
Author(s):  
B. A. Kupershmidt
Keyword(s):  
Calculus Of Variations ◽  
Integrable Systems ◽  
Space Dimension ◽  
Differential Operators ◽  
Differential Forms ◽  
Algebraic Model ◽  
Modern Theory ◽  
Algebraic Models ◽  
Pseudo Differential Operators

The modern theory of integrable systems rests on two fundamental pillars: the classification of Lax [13] and zero-curvature equations [14, 1, 2]; and algebraic models of the classical calculus of variations [9,5] specialized to the residue calculus in modules of differential forms over rings of matrix pseudo-differential operators [9, 6]. Both these aspects of the theory are by now very well understood for integrable systems in one space dimension.

Multivariable Burchnall–Chaundy theory

2007 ◽  
Vol 366 (1867) ◽  
pp. 1155-1177 ◽  
Author(s):  
Emma Previato
Keyword(s):  
Galois Theory ◽  
Differential Operators ◽  
Pseudodifferential Operators ◽  
Geometric Interpretation ◽  
Local Fields ◽  
Differential Galois Theory ◽  
Current State ◽  
Several Variables ◽  
Commutative Subalgebras ◽  
Type Hierarchy

Burchnall & Chaundy (Burchnall & Chaundy 1928 Proc. R. Soc. A 118 , 557–583) classified the (rank 1) commutative subalgebras of the algebra of ordinary differential operators. To date, there is no such result for several variables. This paper presents the problem and the current state of the knowledge, together with an interpretation in differential Galois theory. It is known that the spectral variety of a multivariable commutative ring will not be associated to a KP-type hierarchy of deformations, but examples of related integrable equations were produced and are reviewed. Moreover, such an algebro-geometric interpretation is made to fit into A.N. Parshin's newer theory of commuting rings of partial pseudodifferential operators and KP-type hierarchies which uses higher local fields.

scholarly journals Folding of Hitchin Systems and Crepant Resolutions

2021 ◽  
Author(s):  
Florian Beck ◽  
Ron Donagi ◽  
Katrin Wendland
Keyword(s):  
Fixed Point ◽  
Lie Algebras ◽  
Integrable Systems ◽  
Trace Back ◽  
Dynkin Diagrams ◽  
Or Groups ◽  
Hitchin Systems ◽  
Singular Fibers ◽  
Crepant Resolutions ◽  
Intermediate Jacobian

Abstract Folding of ADE-Dynkin diagrams according to graph automorphisms yields irreducible Dynkin diagrams of $\textrm{ABCDEFG}$-types. This folding procedure allows to trace back the properties of the corresponding simple Lie algebras or groups to those of $\textrm{ADE}$-type. In this article, we implement the techniques of folding by graph automorphisms for Hitchin integrable systems. We show that the fixed point loci of these automorphisms are isomorphic as algebraic integrable systems to the Hitchin systems of the folded groups away from singular fibers. The latter Hitchin systems are isomorphic to the intermediate Jacobian fibrations of Calabi–Yau orbifold stacks constructed by the 1st author. We construct simultaneous crepant resolutions of the associated singular quasi-projective Calabi–Yau three-folds and compare the resulting intermediate Jacobian fibrations to the corresponding Hitchin systems.

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