Abstract
In this paper we show that a semi-commutative Galois extension of associative unital algebra by means of an element \(\tau \), which satisfies \(\tau ^N={\mathbbm {1}}\) (\({\mathbbm {1}}\) is the identity element of an algebra and \(N\ge 2\) is an integer) induces a structure of graded q-differential algebra, where q is a primitive Nth root of unity. A graded q-differential algebra with differential d, which satisfies \(d^N=0, N\ge 2\), can be viewed as a generalization of graded differential algebra. The subalgebra of elements of degree zero and the subspace of elements of degree one of a graded q-differential algebra together with a differential d can be considered as a first order noncommutative differential calculus. In this paper we assume that we are given a semi-commutative Galois extension of associative unital algebra, then we show how one can construct the graded q-differential algebra and when this algebra is constructed we study its first order noncommutative differential calculus. We also study the subspaces of graded q-differential algebra of degree greater than one which we call the higher order noncommutative differential calculus induced by a semi-commutative Galois extension of associative unital algebra. We also study the subspaces of graded q-differential algebra of degree greater than one which we call the higher order noncommutative differential calculus induced by a semi-commutative Galois extension of associative unital algebra. Finally we show that a reduced quantum plane can be viewed as a semi-commutative Galois extension of a fractional one-dimensional space and we apply the noncommutative differential calculus developed in the previous sections to a reduced quantum plane.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abramov, V.: On a graded \(q\)-differential algebra. J. Nonlinear Math. Phys. 13, 1–8 (2006)
Abramov, V.: Algebra forms with \(d^{N}=0\) on quantum plane. Generalized Clifford algebra approach. Adv. Appl. Clifford Algebr. 17, 577–588 (2007)
Abramov, V., Kerner, R.: Exterior differentials of higher order and their covariant generalization. J. Math. Phys. 41(8), 5598–5614 (2000)
Abramov, V., Liivapuu, O.: Connection on module over a graded \(q\)-differential algebra. J. Gen. Lie Theory Appl. 3(2), 112–116 (2008)
Abramov, V., Liivapuu, O.: Generalization of connection on the concept of graded \(q\)-differential algebra. Proc. Estonian Acad. Sci. 59(4), 256–264 (2010)
Abramov, V., Kerner, R., Le Roy, B.: Hypersymmetry: a \(\mathbb{Z}_3\)-graded generalization of supersymmetry. J. Math. Phys. 38, 1650–1669 (1997)
Borowiec, A., Kharchenko, V.K.: Algebraic approach to calculus with partial derivatives. Sib. Adv. Math. 5(2), 10–37 (1995)
Coquereaux, R., Garcia, A.O., Trinchero, R.: Differential calculus and connection on a quantum plane at a cubic root of unity. Rev. Math. Phys. 12(02), 227–285 (2000)
Dubois-Violette, M., Kerner, R.: Universal \(q\) differential calculus and \(q\) analog of homological algebra. Acta Math. Univ. Comenian. 65, 175–188 (1996)
Kapranov, M.: On the q-analog of homological algebra. Preprint Cornell University. arXiv:q-alg/9611005
Kerner, R., Abramov, V.: On certain realizations of \(q\)-deformed exterior differential calculus. Rep. Math. Phys. 43(1–2), 179–194 (1999)
Kerner, R., Suzuki, O.: Internal symmetry groups of cubic algebras. Int. J. Geom. Methods Mod. Phys. 09(6) (2012). doi:10.1142/S0219887812610075
Lawrynowicz, J., Nouno, K., Nagayama, D., Suzuki, O.: A method of noncommutative Galois theory for binary and ternary Clifford analysis. In: Sivasundaram, S. (ed.), 9th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences: ICNPAA 2012, AIP Conference Proceedings 1493, 1007–1014 (2012)
Lawrynowicz, J., Nôno, K., Nagayama, D., Suzuki, O.: A method of noncommutative Galois theory for construction of quark models (Kobayashi-Masukawa Model) I. Bulletin de la Société des Science et des Lettres de Łódź. LXIII, 95–112 (2013)
Trovon, A.: Noncommutative Galois extensions and ternary Clifford analysis. Advances in Applied Clifford Algebras. (to be published)
Acknowledgement
The authors is gratefully acknowledge the Estonian Science Foundation for financial support of this work under the Research Grant No. ETF9328. This research was also supported by institutional research funding IUT20-57 of the Estonian Ministry of Education and Research. The second author also acknowledges his gratitude to the Doctoral School in Mathematics and Statistics for financial support of his doctoral studies at the Institute of Mathematics, University of Tartu. The authors are also grateful for partial support from Linda Peeters Foundation for cooperation between Sweden and Estonia provided by Swedish Mathematical Society.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Abramov, V., Raknuzzaman, M. (2016). Semi-commutative Galois Extension and Reduced Quantum Plane. In: Silvestrov, S., Rančić, M. (eds) Engineering Mathematics II. Springer Proceedings in Mathematics & Statistics, vol 179. Springer, Cham. https://doi.org/10.1007/978-3-319-42105-6_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-42105-6_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-42104-9
Online ISBN: 978-3-319-42105-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)