Differential calculus on almost commutative algebras and applications to the quantum hyperplane.(English)Zbl 1110.81111

Differential graded (DG) $$\rho$$-algebras are defined as a generalization of DG algebras and DG superalgebras. The Lie operation on DG algebras is generalized to the notion of the Lie operation on DG $$\rho$$-algebras. Two examples of DG $$\rho$$-algebras are given: the algebra of forms $$\Omega (A)$$ of an almost commutative algebra $$A$$ and the algebra of noncommutative differential forms $$\Omega _\alpha (A)$$ of a $$\rho$$-algebra $$A$$. Linear connections on a $$\rho$$-bimodule $$M$$ over a $$\rho$$-algebra $$A$$ are presented and extended to the space of forms from $$A$$ to $$M$$. These notions are applied to the quantum hyperplane.

MSC:

 81R60 Noncommutative geometry in quantum theory 16W50 Graded rings and modules (associative rings and algebras) 58C50 Analysis on supermanifolds or graded manifolds

Keywords:

noncommutative geometry
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