91 research outputs found
Calculus and Quantizations over Hopf algebras
In this paper we outline an approach to calculus over quasitriangular Hopf
algebras. We study differential operators in the framework of monoidal
categories equipped with a braiding or symmetry. To be more concrete, we choose
as an example the category of modules over quasitriangular Hopf algebra. We
introduce braided differential operators in a pure algebraic manner.This gives
us a possibility to develop calculus in an intrinsic way without enforcing any
type of Leibniz rule. A general notion of quantization in monoidal categories,
proposed in this paper, is a natural isomorphism of the tensor product
bifunctor equipped with some natural coherence conditions. The quantization
"deforms" all algebraic and differential objects in the monoidal category. We
suggest two ways for calculation of quantizations. One of them reduces the
calculation to non-linear cohomologies. THe other describes quantizations in
terms of multiplicative Hochschild cohomologies of the Grothendieck ring of the
given monoidal category. These constructions are illustrated by some examples.Comment: 46 pages,AMSTEX 2.
Compatibility, multi-brackets and integrability of systems of PDEs
We establish an efficient compatibility criterion for a system of generalized
complete intersection type in terms of certain multi-brackets of differential
operators. These multi-brackets generalize the higher Jacobi-Mayer brackets,
important in the study of evolutionary equations and the integrability problem.
We also calculate Spencer delta-cohomology of generalized complete
intersections and evaluate the formal functional dimension of the solutions
space. The results are applied to establish new integration methods and solve
several differential-geometric problems.Comment: Some modifications in sections 6.1-2; new references're adde
Dimension of the solutions space of PDEs
We discuss the dimensional characterization of the solutions space of a
formally integrable system of partial differential equations and provide
certain formulas for calculations of these dimensional quantities.Comment: Contribution to the conference GIFT-200
The categorical theory of relations and quantizations
In this paper we develope a categorical theory of relations and use this
formulation to define the notion of quantization for relations. Categories of
relations are defined in the context of symmetric monoidal categories. They are
shown to be symmetric monoidal categories in their own right and are found to
be isomorphic to certain categories of bicomodules. Properties of
relations are defined in terms of the symmetric monoidal structure. Equivalence
relations are shown to be commutative monoids in the category of relations.
Quantization in our view is a property of functors between monoidal categories.
This notion of quantization induce a deformation of all algebraic structures in
the category, in particular the ones defining properties of relations like
transitivity and symmetry.Comment: corrected typo
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