arXiv subjects

Th. Th. Voronov

Publications and source records attributed to Th. Th. Voronov.

9 recordsLinked to original sources

arXiv2024

On the Buchstaber--Rees theory of "Frobenius $n$-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.

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arXiv2014

Second order operators on the algebra of densities and a groupoid of connections

We consider the geometry of second order linear operators acting on the commutative algebra of densities on a (super)manifold introduced in our previous work. In the conventional language, operators on the algebra of densities correspond to operator pencils. This algebra has a natural invariant scalar product. We consider self-adjoint operators on the algebra of densities and analyze the corresponding "canonical operator pencils" passing through a given operator on densities of a particular weight. There are singular values for the pencil parameters. This leads to an interesting geometrical picture. In particular we obtain operators that depend on equivalence classes of connections and we study a groupoid of connections such that the orbits of this groupoid are these equivalence classes. Based on this point of view we analyze two examples: the second order canonical operator on an odd symplectic supermanifold appearing in the Batalin-Vilkovisky geometry and the Sturm-Liouville operator on the line related with classical constructions of projective geometry. We also consider a canonical second order semidensity arising on odd symplectic supermanifolds, which has some resemblance with mean curvature in Riemannian geometry.

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arXiv2013

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 Frölicher--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.

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arXiv2012

Geometry of differential operators of second order, the algebra of densities, and groupoids

In our previous works, we introduced, for each (super)manifold, a commutative algebra of densities. It is endowed with a natural invariant scalar product. In this paper, we study geometry of differential operators of second order on this algebra. In the more conventional language they correspond to certain operator pencils. We consider the self-adjoint operators and analyze the operator pencils that pass through a given operator acting on densities of a particular weight. There are `singular values' for pencil parameters. They are related with interesting geometric picture. In particular, we obtain operators that depend on certain equivalence classes of connections (instead of connections as such). We study the corresponding groupoids. From this point of view we analyze two examples: the canonical Laplacian on an odd symplectic supermanifold appearing in Batalin--Vilkovisky geometry and the Sturm--Liouville operator on the line, related with classical constructions of projective geometry. We also consider the canonical second order semi-density arising on odd symplectic supermanifolds, which has some similarity with mean curvature of surfaces in Riemannian geometry.

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arXiv2010

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.

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arXiv2008

Higher Poisson Brackets and Differential Forms

We show how the relation between Poisson brackets and symplectic forms can be extended to the case of inhomogeneous multivector fields and inhomogeneous differential forms (or pseudodifferential forms). In particular we arrive at a notion which is a generalization of a symplectic structure and gives rise to higher Poisson brackets. We also obtain a construction of Koszul type brackets in this setting.

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arXiv2007

Operators on superspaces and generalizations of the Gelfand-Kolmogorov theorem

(Write-up of a talk at the Bialowieza meeting, July 2007.) Gelfand and Kolmogorov in 1939 proved 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 recently extended this to all symmetric powers $\Sym^n(X)$ using their notion of the Frobenius $n$-homomorphisms. We give a simplification and a further extension of this theory, which is based, rather unexpectedly, on results from super linear lgebra.

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arXiv2007

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.

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arXiv2004

Berezinians, Exterior Powers and Recurrent Sequences

We study power expansions of the characteristic function of a linear operator $A$ in a $p|q$-dimensional superspace $V$. We show that traces of exterior powers of $A$ satisfy universal recurrence relations of period $q$. `Underlying' recurrence relations hold in the Grothendieck ring of representations of $\GL(V)$. They are expressed by vanishing of certain Hankel determinants of order $q+1$ in this ring, which generalizes the vanishing of sufficiently high exterior powers of an ordinary vector space. In particular, this allows to explicitly express the Berezinian of an operator as a rational function of traces. We analyze the Cayley--Hamilton identity in a superspace. Using the geometric meaning of the Berezinian we also give a simple formulation of the analog of Cramer's rule.

math.DG↗