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

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*Catalin Ciupala*

**Address.**

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

**E-mail. **cciupala@yahoo.com

**Abstract.**

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.