nLab Buchstaber–Rees theorem

under construction

Idea

Gelfand and Kolmogorov proved in 1939 that a compact Hausdorff topological space $X$ can be canonically embedded into the infinite‐dimensional vector space $C(X)^*$, the dual space of the algebra of continuous functions $C(X)$, as an “algebraic variety”, specified by an infinite system of quadratic equations.

Buchstaber and Rees have extended this to all symmetric powers $Sym_n(X)$ using their notion of the Frobenius $n$‐homomorphisms.

Related concepts

• Gelfand-Kolmogorov theorem

References

• Victor M. Buchstaber, Elmer G. Rees, The Gelfand map and symmetric products, Selecta Math. 8 (2002), no. 4, 523–535; …, Russian Math. Surveys 59 (1(355)):125–145, 2004; Frobenius n-homomorphisms, transfers and branched coverings, Math. Proc. Cambr. Phil. Soc. 2007, arXiv:math.RA/0608120 .

• I. M. Gel’fand, A. N. Kolmogorov. Dokl. Akad. Nauk SSSR 22: 11–15, 1939.

• H. M. Khudaverdian, Th. Th. Voronov, dadada, Lett. Math. Phys., 74(2):201–228, 2005; Operators on superspaces and generalizations of the Gelfand–Kolmogorov theorem, AIP Conf. Proc. 956, 149 (2007);doi; On generalized symmetric powers and a generalization of Gel’fand-Kolmogorov-Buchstaber-Rees theorem, pdf; A short proof of the Buchstaber–Rees theorem, Phil. Trans. R. Soc. A 369, no. 1939, 1334-1345 (2011) full, pdf (keywords: Berezinian (superdeterminant), Frobenius recursion, symmetric powers, maps of algebras, $n$-homomorphism, $p|q$-homomorphism

• Alexander Pavlov, Evgenij Troitsky, Quantization of branched coverings, Russian J. of Math. Phys. 18, N 3 (2011), 338-352 arxiv/1002.3491