Differential calculus on almost commutative algebras and applications to the quantum hyperplane

Catalin Ciupala

Department of Differential Equations, Faculty of Mathematics and Informatics, University Transilvania of Brasov, 2200 Brasov, Romania

E-mail. cciupala@yahoo.com

In this paper we introduce a~new class of differential graded algebras named DG $\rho $-algebras and present Lie operations on this kind of algebras. We give two examples: the algebra of forms and the algebra of noncommutative differential forms of a~$\rho $-algebra. Then we introduce linear connections on a~$\rho $-bimodule $M$ over a~$\rho $-algebra~$A$ and extend these connections to the space of forms from $A$ to $M$. We apply these notions to the quantum hyperplane.

AMSclassification. 81R60, 16W99, 53C04.

Keywords. Noncommutative geometry, almost commutative algebra, linear connections, quantum hyperplane.