On generalized symmetric powers and a generalization of Kolmogorov-Gelfand-Buchstaber-Rees theory

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.Comment: 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 theore

Operator pencils on the algebra of densities

In this paper we continue to study equivariant pencil liftings and differential operators on the algebra of densities. We emphasize the role that the geometry of the extended manifold plays. Firstly we consider basic examples. We give a projective line of diff($M$)-equivariant pencil liftings for first order operators, and the canonical second order self-adjoint lifting. Secondly we study pencil liftings equivariant with respect to volume preserving transformations. This helps to understand the role of self-adjointness for the canonical pencils. Then we introduce the Duval-Lecomte-Ovsienko (DLO)-pencil lifting which is derived from the full symbol calculus of projective quantisation. We use the DLO-pencil lifting to describe all regular proj-equivariant pencil liftings. In particular the comparison of these pencils with the canonical pencil for second order operators leads to objects related to the Schwarzian. Within this paper the question of whether the pencil lifting factors through a full symbol map naturally arises.Comment: 23 pages, LaTeX file Small corrections are mad

A short proof of the Buchstaber-Rees theorem

We give a short proof of the Buchstaber-Rees theorem concerning symmetric powers. The proof is based on the notion of a formal characteristic function of a linear map of algebras.Comment: 11 pages. LaTeX2

Geometric constructions on the algebra of densities

The algebra of densities \Den(M) is a commutative algebra canonically associated with a given manifold or supermanifold $M$. We introduced this algebra earlier in connection with our studies of Batalin--Vilkovisky geometry. The algebra \Den(M) is graded by real numbers and possesses a natural invariant scalar product. This leads to important geometric consequences and applications to geometric constructions on the original manifold. In particular, there is a classification theorem for derivations of the algebra \Den(M). It allows a natural definition of bracket operations on vector densities of various weights on a (super)manifold $M$, similar to how the classical Fr\"{o}licher--Nijenhuis theorem on derivations of the algebra of differential forms leads to the Nijenhuis bracket. It is possible to extend this classification from "vector fields" (derivations) on \Den(M) to "multivector fields". This leads to the striking result that an arbitrary even Poisson structure on $M$ possesses a canonical lifting to the algebra of densities. (The latter two statements were obtained by our student A.Biggs.) This is in sharp contrast with the previously studied case of an odd Poisson structure, where extra data are required for such a lifting.Comment: LaTeX, 23 p

Operator pencil passing through a given operator

Let $\Delta$ be a linear differential operator acting on the space of densities of a given weight \lo on a manifold $M$. One can consider a pencil of operators \hPi(\Delta)=\{\Delta_\l\} passing through the operator $\Delta$ such that any \Delta_\l is a linear differential operator acting on densities of weight \l. This pencil can be identified with a linear differential operator \hD acting on the algebra of densities of all weights. The existence of an invariant scalar product in the algebra of densities implies a natural decomposition of operators, i.e. pencils of self-adjoint and anti-self-adjoint operators. We study lifting maps that are on one hand equivariant with respect to divergenceless vector fields, and, on the other hand, with values in self-adjoint or anti-self-adjoint operators. In particular we analyze the relation between these two concepts, and apply it to the study of \diff(M)-equivariant liftings. Finally we briefly consider the case of liftings equivariant with respect to the algebra of projective transformations and describe all regular self-adjoint and anti-self-adjoint liftings.Comment: 32 pages, LaTeX fil

On the Buchstaber--Rees theory of "Frobenius nn-homomorphisms" and its generalization

This is a survey of our results on the theory of $n$-homomorphisms of Buchstaber--Rees and its generalization that we obtained. In short, we are concerned with classes of linear maps between commutative rings that can be described the "next level after ring homomorphisms" with respect to multiplicative properties. Our main tool is a construction which we call the "characteristic function" -- whose functional properties encode algebraic properties of a linear map in question. Namely, if the characteristic function is polynomial of degree $n$, the map is an $n$-homomorphism in the sense of Buchstaber--Rees, and our approach simplifies their theory substantially. If the characteristic function is an irreducible rational fraction with the numerator and denominator of degrees $p$ and $q$ respectively, we arrive at a new notion of a "$p|q$-homomorphism". Examples of $p|q$-homomorphisms are sums and differences of ring homomorphisms. Our construction is motivated by our earlier results in superalgebra/supergeometry concerning Berezinians and super exterior powers.Comment: LaTeX, 10p