Abstract
Given a graded associative unital algebra we construct a graded \(q\)-differential algebra, where \(q\) is a primitive \(N\)th root of unity and prove that the generalized cohomologies of the corresponding \(N\)-complex are trivial. We construct a graded \(q\)-differential algebra of polynomials and introduce a notion of connection form. We find explicit formula for the curvature of connection form and prove Bianchi identity.
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Acknowledgments
The authors are grateful to Michel Dubois-Violette of the University Paris XI for valuable discussions and suggestions that improved the manuscript. The authors also gratefully acknowledge the financial support of the Estonian Science Foundation under the research grant ETF9328, target finance grant SF0180039s08 and ESF DoRa programme.
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Abramov, V., Liivapuu, O. (2014). Graded \(q\)-Differential Polynomial Algebra of Connection Form. In: Makhlouf, A., Paal, E., Silvestrov, S., Stolin, A. (eds) Algebra, Geometry and Mathematical Physics. Springer Proceedings in Mathematics & Statistics, vol 85. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55361-5_21
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DOI: https://doi.org/10.1007/978-3-642-55361-5_21
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