Abstract
In this article we construct a cochain complex of a complex Clifford algebra with coefficients in itself in a combinatorial fashion and we call the corresponding cohomology by Clifford cohomology. We show that Clifford cohomology controls the deformation of a complex Clifford algebra and can classify them up to Morita equivalence. We also study Hochschild cohomology groups and formal deformations of the algebra of smooth sections of a complex Clifford algebra bundle over an even dimensional orientable Riemannian manifold M which admits a \(Spin^{c}\) structure.
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References
Bursztyn, H., Waldmann, S.: Classifying Morita Equivalent Star Products, Clay Mathematics Proceedings, vol. 16. Topics in Noncommutative Geometry, Third Luis Santaló Winter School-CIMPA Research School, Topics in Noncommutative Geometry, Universidad de Buenos Aires, Buenos Aires, Argentina (2010)
Doubek, M., Markl, M., Zima, P.: Deformation Theory (Lecture Notes). arXiv: 0705.3719v3 [math.AG] (2009)
Gerstenhaber, M., Schack, D.S.: Algebraic cohomology and deformation theory. In: Hazewinkel, M., Gerstenhaber, M. (eds.) Deformation Theory of Algebras and Structures and Applications. NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 247. Kluwer Academic Publishers Group, Dordrecht (1988)
Gerstenhaber, M.: On the deformation of rings and algebras. Ann. Math. (2) 79, 59–103 (1964)
Kassel, C.: Homology and Cohomology of Associative Algebras—A Concise Introduction to Cyclic Homology. École thématique. Ao\(\hat{{\rm u}}\)t 2004 à ICTP, Trieste (Italy) (2006). cel- 00119891
Kontsevich, M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 56, 271–294 (2003)
Lam, T.Y.: Lectures on Modules and Rings. Graduate Texts in Mathematics. Springer, Berlin (1999)
Lawson, H.B., Michelshon, M.-L.: Spin Geometry. Princeton University Press, Princeton (1990)
Loday, J.-L.: Cyclic Homology. Grundlehren der Mathematischen Wissenschaften, vol. 301. Springer, Berlin (1992)
Lounesto, P.: Clifford Algebras and Spinors. Cambridge University Press, Cambridge (1997)
Lundholm, D., Svensson, L.: Clifford Algebra, Geometric Algebra and Applications, arXiv: 0907.5356v1 [mathph] (2009)
Nestruev, J.: Smooth Manifolds and Observables. Graduate Texts in Mathematics, vol. 220. Springer, Berlin (2003)
Waldmann, S.: Poisson-Geometrie und Deformationsquantisierung: Eine Einführung, Springer-Lehrbuch Masterclass. Springer, Heidelberg (2007)
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The authors thank Professor Stefan Waldmann for his valuable suggestions and comments.
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Communicated by Vladimír Soucek.
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Banerjee, B., Mukherjee, G. A Note on Cohomology of Clifford Algebras. Adv. Appl. Clifford Algebras 34, 16 (2024). https://doi.org/10.1007/s00006-024-01324-z
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DOI: https://doi.org/10.1007/s00006-024-01324-z