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- Title
COHOMOLOGIES, EXTENSIONS AND DEFORMATIONS OF DIFFERENTIAL ALGEBRAS OF ARBITRARY WEIGHT.
- Authors
LI GUO; YUNNAN LI; YUNHE SHENG; GUODONG ZHOU
- Abstract
As an algebraic structure underlying the differential calculus and differential equations, a differential algebra is an associative algebra equipped with a linear map satisfying the Leibniz rule. The subject has been studied for about a century and has become an important area of mathematics. In recent years the area has been expanded to the noncommutative associative and Lie algebra contexts and to the case when the defining operator identity has a weight in order to include difference operators. This paper provides a cohomology theory for differential algebras of arbitrary weight, via a uniform approach to cover both the zero weight case which is similar to the earlier study of differential Lie algebras, and the non-zero weight case which poses challenges. The cohomology of a differential algebra is related to the Hochschild cohomology by a type of long exact sequence for relative homology. As an application, abelian extensions of a differential algebra are classified by the second cohomology group. Furthermore, formal deformations of a differential algebra are characterized by the second cohomology group and the rigidity of a differential algebra is characterized by the vanishing of the second cohomology group.
- Subjects
COHOMOLOGY theory; ASSOCIATIVE algebras; DIFFERENTIAL calculus; LIE algebras; DIFFERENCE operators; LINEAR operators; UNIFORM algebras; DIFFERENTIAL algebra
- Publication
Theory & Applications of Categories, 2022, Vol 38, p1409
- ISSN
1201-561X
- Publication type
Article