Abstract
We consider the algebra k[X 2,XY,Y 2] where characteristic of the field k is zero. We compute a differential calculus, introduced earlier by the authors, by associating an algebraic spectral triple with this algebra. This algebra can also be viewed as the coordinate ring of the singular variety UV–W 2 and hence, is a quadratic algebra. We associate two canonical algebraic spectral triples with this algebra and its quadratic dual, and compute the associated Connes’ calculus. We observe that the resulting Connes’ calculi are also quadratic algebras, and they turn out to be quadratic dual to each other.
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References
T. Brzeziński and S. Majid, A class of bicovariant differential calculi on Hopf algebras, Lett. Math. Phys., 26(1) (1992), 67–78.
P. S. Chakraborty and S. Guin, Multiplicativity of Connes’ calculus, http://arxiv.org/abs/1402.5735, arXiv:1402.5735 [math.QA] (submitted).
A. Connes, Noncommutative Geometry, Academic Press, Inc., San Diego, CA, 1994.
M. Dubois-Violette, Lectures on graded differential algebras and noncommutative geometry, In: Y. Maeda, et al. (Eds.), Noncommutative Differential Geometry and its Applications to Physics, Kluwer Academic Publishers, 2001, pp. 245–306. Shonan, Japan, 1999.
D. Kastler and D. Testard, Quantum forms of tensor products, Comm. Math. Phys., 155(1) (1993), 135–142.
J. Madore, An introduction to noncommutative differential geometry and its physical applications, Second edition, London Mathematical Society Lecture Note Series, 257, Cambridge University Press, Cambridge, 1999.
S. Majid, Classification of bicovariant differential calculi, J. Geom. Phys., 25(1-2) (1998), 119–140.
A. Polishchuk and L. Positselski, Quadratic algebras, University Lecture Series, 37, American Mathematical Society, Providence, RI, 2005.
P. Podleś, Differential calculus on quantum spheres, Lett. Math. Phys., 18(2) (1989), 107–119.
S. L. Woronowicz, Differential calculus on compact matrix pseudogroups (quantum groups), Comm. Math. Phys., 122(1) (1989), 125–170.
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Dedicated to Prof. Kalyan B. Sinha on occasion of his 70th birthday.
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Chakraborty, P.S., Guin, S. Noncommutative differential calculus on a quadratic algebra. Indian J Pure Appl Math 46, 495–515 (2015). https://doi.org/10.1007/s13226-015-0149-0
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DOI: https://doi.org/10.1007/s13226-015-0149-0