Integrable graded manifolds and nonlinear equations
type: Item
instance of: scholarly article
title: Integrable graded manifolds and nonlinear equations
zbMATH Open document ID: 0588.17004
DOI: 10.1007/BF01468141
published in: Communications in Mathematical Physics
publication date: 1984-01-01T00:00:00Z
review text: The correspondence between integrable embeddings of 2-manifolds in Lie algebra spaces and coefficient series of certain Lie algebra elements is discussed and, moreover, its generalizations in the case of such a 2, 2- supermanifold and Lie superalgebras are considered. The method is the algebraic analysis of graded Lie algebra commutation relations. In the ''non-super problem'' the vanishing of the curvature tensor of the flat enveloping space leads to Dynkin's procedure for integral embeddings of sl(2)-like subalgebras in simple Lie algebras. The relevant equations may be sometimes explicitly solved by means of the classical Goursat problem. In a special case these equations turn out to be the Liouville and the sine-Gordon equations as shown in appendix B. In the ''super'' problem the author considers a generalized procedure to the above one whose correctness may be proved if the enveloping space is equipped with the structure of \(sl(n,n+1)\) superalgebras. Then the case \(n=1\) corresponds to the supersymmetrical Liouville and sine-Gordon equations.
Mathematics Subject Classification ID: 17A70
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 35Q99
zbMATH DE Number: 3944089
zbMATH Keywords: supermanifolds
zbMATH Keywords: supersymmetrical Liouville equation
zbMATH Keywords: sine-Gordon equation
zbMATH Keywords: Toda lattice
zbMATH Keywords: Cartan element
zbMATH Keywords: integrable embeddings
zbMATH Keywords: Lie superalgebras
author: Mikhail V. Saveliev
MaRDI profile type: Publication
cites work: Q3871494
cites work: Q3897665
cites work: Q3681611
cites work: Representation theory and integration of nonlinear spherically symmetric equations to gauge theories
cites work: Two-dimensional exactly and completely integrable dynamical systems
cites work: Q4741063
cites work: To a transformation theory of two-dimensional integrable systems
cites work: On reduction of a two-dimensional generalized Toda lattice to an ordinary differential equations system
cites work: Q5813959
cites work: Q5580319
cites work: SIMPLE IRREDUCIBLE GRADED LIE ALGEBRAS OF FINITE GROWTH
cites work: Q4130629
cites work: Q3043319
cites work: Q3733819
cites work: Infinite-dimensional algebras, Dedekind's \(\eta\)-function, classical Möbius function and the very strange formula
cites work: Q3744333
cites work: Lie superalgebras
cites work: Supersymmetric two-dimensional Toda lattice
cites work: Q4747274
cites work: General solutions of nonlinear equations in the geometric theory of the relativistic string
description: scientific article; zbMATH DE number 3944089