On Generalized Symmetric Powers and a Generalization of Kolmogorov–gelfand–buchstaber–rees Theory

In 1939, Gelfand and Kolmogorov published a paper [4], where they showed that for a (compact Hausdorff) topological space X, homomorphisms of the algebra of continuous functions C(X) to numbers are in a one-to-one correspondence with points of X. Paper [4] is famous for giving birth to the theory of normed rings. However, it is worth stressing that the Gelfand–Kolmogorov theorem may be viewed as a description of the image of the canonical embedding of X into the infinite-dimensional linear space V = A, where A = C(X), by a system of quadratic equations f(1) = 1, f(a) − f(a) = 0, indexed by elements of A. This aspect was recently emphasized by Buchstaber and Rees (see [1] and references therein), who showed that there is a natural embedding into V not only for X, but also for all its symmetric powers SymX. To this end, algebra homomorphisms should be replaced by the so-called ‘n-homomorphisms’, and quadratic equations describing the image, by certain algebraic equations of higher degree. Buchstaber and Rees’ theory was motivated by their earlier study of Hopf objects for multi-valued groups. (In the hindsight, the notion of an n-homomorphism of algebras was present, implicitly, in Frobenius’s notion of higher group characters.) In the present note we give a further natural generalization of Buchstaber–Rees’s theory. Namely, for a set or topological space X there is a functorial object SymX, p, q > 0, and for a commutative algebra with unit A, a corresponding algebra SA (see definitions below). We call them ‘generalized symmetric powers’. There is a canonical embedding of SymX into V = A, and on the level of algebras, there is a description of algebra homomorphisms SA → B. This is done in terms of the new notion of a ‘p|q-homomorphism’. Our work was substantially motivated by the results in super geometry in [3], from which comes our main tool, the ‘characteristic function’ of a linear map of algebras. Let A and B be commutative associative algebras with unit. Consider an arbitrary linear map f : A → B. Its characteristic function is defined to be R(f , a, z) = e , where a ∈ A and z is a formal parameter. Example: if f is an algebra homomorphism, then R(f , a, z) = 1+ f(a)z, i.e., a linear polynomial. Algebraic properties of