35 research outputs found
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()-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
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 . 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 , 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 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
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
Operator pencil passing through a given operator
Let be a linear differential operator acting on the space of
densities of a given weight \lo on a manifold . One can consider a pencil
of operators \hPi(\Delta)=\{\Delta_\l\} passing through the operator
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 -homomorphisms" and its generalization
This is a survey of our results on the theory of -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 , the map is an -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 and respectively, we arrive at a
new notion of a "-homomorphism". Examples of -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