Integrable graded manifolds and nonlinear equations
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.
scientific article; zbMATH DE number 3944089
scientific article