On generalized symmetric powers and a generalization of Kolmogorov-Gelfand-Buchstaber-Rees theory
Abstract
The classical Kolmogorov-Gelfand theorem gives an embedding of a (compact Hausdorff) topological space X into the linear space of all linear functionals C(X)^* on the algebra of continuous functions C(X). The image is specified by algebraic equations: f(ab)=f(a)f(b) for all functions a, b on X; that is, the image consists of all algebra homomorphisms of C(X) to numbers. Buchstaber and Rees have found that not only X, but all symmetric powers of X can be embedded into the space C(X)^*. The embedding is again given by algebraic equations, but more complicated. Algebra homomorphisms are replaced by the so-called "n-homomorphisms", the notion that can be traced back to Frobenius, but which explicitly appeared in Buchstaber and Rees's works on multivalued groups. We give a further natural generalization of Kolmogorov-Gelfand-Buchstaber-Rees theory. Symmetric powers of a space X or of an algebra A are replaced by certain "generalized symmetric powers" Sym^{p|q}(X) and S^{p|q}A, which we introduce, and n-homomorphisms, by the new notion of "p|q-homomorphisms". Important tool of our study is a certain "characteristic function" R(f,a,z), which we introduce for an arbitrary linear map of algebras f, and whose functional properties with respect to the variable z reflect algebraic properties of the map f.
Keywords
Cite
@article{arxiv.math/0612072,
title = {On generalized symmetric powers and a generalization of Kolmogorov-Gelfand-Buchstaber-Rees theory},
author = {H. M. Khudaverdian and Th. Th. Voronov},
journal= {arXiv preprint arXiv:math/0612072},
year = {2019}
}
Comments
LaTeX, 7 pages (3+4). In this new version we slightly edited the main text, and added to it an Appendix giving details of some constructions and a short direct proof of Buchstaber--Rees's main theorem