Related papers: Toward quantized Picard-Vessiot theory, Further ob…
In the present paper, we will show that three apparently disjoint objects: Galois representations arising from twenty-seven lines on a cubic surface (number theory and arithmetic algebraic geometry), Picard modular forms (automorphic…
We introduce the parameterized generic Galois group of a q-difference module, that is a differential group in the sense of Kolchin. It is associated to the smallest differential tannakian category generated by the q-difference module,…
Inspired by the commutator and anticommutator algebras derived from algebras graded by groups, we introduce noncommutatively graded algebras. We generalize various classical graded results to the noncommutatively graded situation concerning…
We consider some examples of superintegrable system which were recently isolated through a differential Galois group analysis. The identity of these systems is clarified and the corresponding Poisson algebras derived.
Much work has gone into matrix representations of Galois groups, but there is a whole new class of naturally occurring representations that have as yet gone almost unnoticed. In fact, it is well-known in various areas of mathematics that…
The aim of this article is to provide a method to prove the irreducibility of non-linear ordinary differential equations by means of the differential Galois group of their variational equations along algebraic solutions. We show that if the…
We consider GLq(N)-covariant quantum algebras with generators satisfying quadratic polynomial relations. We show that, up to some inessential arbitrariness, there are only two kinds of such quantum algebras, namely, the algebras with…
The Galois theory of logarithmic differential equations with respect to relative D-groups in partial differential-algebraic geometry is developed.
We show that if a holomorphic Hamiltonian system is holomorphically integrable in the non-commutative sense in a neighbourhood of a non-equilibrium phase curve which is located at a regular level of the first integrals, then the identity…
We review Hopf-Galois extensions, in particular faithfully flat ones, accepted to be the noncommutative algebraic dual of a principal bundle. We also make a short digression into how quantum groups relate to Hopf-Galois extensions. Several…
We reconsider differential geometry from the point of view of the quantum theory of non-relativistic spinning particles, which provides examples of supersymmetric quantum mechanics. This enables us to encode geometrical structure in…
This paper describes the classification of analytic $q$-difference equations. The difference Galois groups are computed. A tentative description of the universal difference Galois group is given.
We develop a Galois theory of commutative rings under actions of finite inverse semigroups. We present equivalences for the definition of Galois extension as well as a Galois correspondence theorem. We also show how the theory behaves in…
In this article we discuss some general results on the covariant Picard groupoid in the context of differential geometry and interpret the problem of lifting Lie algebra actions to line bundles in the Picard groupoid approach.
The theory of R. Crowell on derived modules is approached within the theory of non-commutative differential modules. We also seek analogies to the theory of cotangent complex from differentials in the commutative ring setting. Finally we…
Picard--Vessiot theorem (1910) provides a necessary and sufficient condition for solvability of linear differential equations of order $n$ by quadratures in terms of its Galois group. It is based on the differential Galois theory and is…
This expository paper presents some applications of the parameterized Picard-Vessiot theory for ordinary differential equations, mainly related to monodromy.
The Vlasov-Poisson equations, fundamental in plasma physics and astrophysical applications, are rendered linear, finite-dimensional, and discrete by second quantization. Conditions for correspondence between the pre-quantized and quantized…
We introduce a novel integrability-preserving discretization for a broad class of differential equations with variable coefficients, encompassing both linear and nonlinear cases. The construction is achieved via a categorical approach that…
The purpose of this paper is to connect two subjects: the theory of quantum integrable systems (complete commutative rings of differential operators), and differential Galois theory. We define quantum completely integrable systems (QCIS),…