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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
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