arXiv subjects
Publications and source records attributed to Valentin Lychagin.
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.
We study the equivalence problem of submanifolds with respect to a transitive pseudogroup action. The corresponding differential invariants are determined via formal theory and lead to the notions of k-variants and k-covariants, even in the case of non-integrable pseudogroup. Their calculation is based on the cohomological machinery: We introduce a complex for covariants, define their cohomology and prove the finiteness theorem. This implies the well-known Lie-Tresse theorem about differential invariants. We also generalize this theorem to the case of pseudogroup action on differential equations.
We generalize the notion of involutivity to systems of differential equations of different orders and show that the classical results due to Guillemin and Quillen relating involutivity, restrictions, characteristics and characteristicity, known for first order systems, extend to the general context, though in a modified form. This involves, in particular, a new definition of strong characteristicity. The proof exploits a spectral sequence relating Spencer delta-cohomology of a symbolic system and its restriction to a non-characteristic subspace.
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 $A-A$ 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.
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.