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.


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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