Mappings of supermanifolds
The paper concerns the paper of \textit{A. S. Shvarts} [ibid. 60, No.1, 37- 42 (1984)], where the concept of superspaces, supermanifolds etc., based on Grassmann algebras and more convenient for physics applications than the standard concept based on sheaf theory, was developed. The author proves that the concept of a smooth mapping between (p,q)-dimensional supermanifolds, defined in the sense of Shvarts, is identical to the concept of a morphism of supermanifolds in the standard approach and hence, the Shvarts' definition of a (p,q)-dimensional supermanifold is equivalent to the standard one. Theorem 1. The smooth mappings of \({\mathbb{R}}^{p,q}\) into \({\mathbb{R}}^{p',q'}\) are in a one-to-one correspondence with the parity- preserving homomorphisms of the Berezin algebra \(B^{B',q'}\) into the algebra \(B^{p,q}\). - A similar correspondence is constructed even for non-smooth mappings.
scientific article; zbMATH DE number 3909372