The Differential Azumaya Algebras and Non-commutative Picard–Vessiot Cocycles
نویسندگان
چکیده
A differential Azumaya algebra, and in particular a differential matrix algebra, over a differential field K with constants C is trivialized by a Picard–Vessiot (differential Galois) extension E. This yields a bijection between isomorphism classes of differential algebras and Picard–Vessiot cocycles Z(G(E/K), PGLn(C)) which cobound in Z (G(E/K), PGLn(E)).
منابع مشابه
Generic Rings for Picard–Vessiot Extensions and Generic Differential Equations
Let G be an observable subgroup of GLn. We produce an extension of differential commutative rings generic for Picard–Vessiot extensions with
متن کاملOn Picard–Vessiot extensions with group PGL3
Let F be a differential field of characteristic zero with algebraically closed field of constants. We provide an explicit description of the twisted Lie algebras of PGL3-equivariant derivations on the coordinate rings of F -irreducible PGL3-torsors in terms of nine-dimensional central simple algebras over F . We use this to construct a Picard–Vessiot extension which is the function field of a n...
متن کاملDifferential Central Simple Algebras and Picard–Vessiot Representations
A differential central simple algebra, and in particular a differential matrix algebra, over a differential field K with constants C can be trivialized by a Picard–Vessiot (differential Galois) extension E. In the matrix algebra case, there is a correspondence between K algebras trivialized by E and representations of the Galois group of E over K in PGLn(C), which can be interpreted as cocyles ...
متن کاملA Functorial Approach for Dynamical Systems over Galois Differential Algebras: Integrability and Picard-vessiot Theory
A categorical theory for the discretization of a large class of dynamical systems with variable coefficients is proposed. It is based on the existence of covariant functors between the Rota category of Galois differential algebras and suitable categories of abstract dynamical systems. The integrable maps obtained share with their continuous counterparts a large class of solutions and, in the li...
متن کاملA Categorical Approach to Picard-vessiot Theory
Picard-Vessiot rings are present in many settings like differential Galois theory, difference Galois theory and Galois theory of Artinian simple module algebras. In this article we set up an abstract framework in which we can prove theorems on existence and uniqueness of Picard-Vessiot rings, as well as on Galois groups corresponding to the Picard-Vessiot rings. As the present approach restrict...
متن کامل