Generalized differential spaces with $d^N=0$ and the $q$-differential calculus

Abstract

We present some results concerning the generalized homologies associated with nilpotent endomorphisms $d$ such that $d^N=0$ for some integer $N\ge 2$. We then introduce the notion of graded $q$-differential algebra and describe some examples. In particular we construct the $q$-analog of the simplicial differential on forms, the $q$-analog of the Hochschild differential and the $q$-analog of the universal differential envelope of an associative unital algebra.

8 pages, Latex2e, uses pb-diagram, available at http://qcd.th.u-psud.fr, to be published in the Proceedings of the 5th Colloquium ``Quantum Groups and Integrable Systems", Prague, June 1996

References in corpus (2)

• On the q-analog of homological algebra
• Universal $q$-differential calculus and $q$-analog of homological algebra

Cited by in corpus (14)

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• Homogeneous algebras
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• A differential calculus on Z${}_3$-graded quantum superspace ${\mathbb{R}}_q(2|1)$
• Left covariant differential calculi on ${\widetilde{\mathrm{GL}}}_q(2)$
• Quantum de Rham complex with $d^3=0$ differential
• Tensor product of N-complexes and generalization of graded differential algebras
• N-Complexes and Higher Spin Gauge Fields
• On the tensor product construction for q-differential algebras
• Right-covariant differential calculus on $\mathcal{F}({\mathbb{C}}_q^{2|1})$
• On the (3,N) Maurer-Cartan equation