Abstract
We show that a semi-commutative Galois extension of a unital associative algebra can be endowed with the structure of a graded q-differential algebra. We study the first and higher order noncommutative differential calculus of semi-commutative Galois extension induced by the graded q-differential algebra. As an example we consider the quaternions which can be viewed as the semi-commutative Galois extension of complex numbers.
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Abramov V., Kerner R.: Exterior differentials of higher order and their covariant generalization. J. Math. Phys. 41(8), 5598–5614 (2000)
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 Algebras 17, 577–588 (2007)
Abramov V., Liivapuu O.: Connection on module over a graded q-differential algebra. J. General. Lie Theory Appl. 3(2), 112–116 (2008)
Abramov V., Liivapuu O.: Generalization of connection on the concept of graded q-differential algebra. Proc. Eston. Acad. Sci. 59(4), 256–264 (2010)
Borowiec A., Kharchenko V.K.: Algebraic approach to calculus with partial derivatives. Siber. Adv. Math. 5(2), 10–37 (1995)
Dubois-Violette M., Kerner R.: Universal q differential calculus and q analog of homological algebra. Acta Math. Univ. Comen. 65, 175–188 (1996)
Kapranov, M.M.: On the q-analog of homological algebra. arXiv:math.QA/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, 1261007 (2012)
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: ICNAAP 2012, AIP Conf. Proc., vol. 1443, p. 1007 (2012)
Lawrynowicz, J., Nôno, K., Nagayama, D., Suzuki, O.: A method of noncommutative Galois theory for construction of quark models (Kobayashi–Masukawa model) I. Bull. Soc.Sci. Lett. Łódź LXIII, 95–112 (2013)
Trovon, A., Suzuki, O.: Noncommutative Galois extensions and ternary Clifford analysis. Adv. Appl. Clifford Algebras (2015). doi:10.1007/s00006-015-0565-6
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Abramov, V. Noncommutative Galois Extension and Graded q-Differential Algebra. Adv. Appl. Clifford Algebras 26, 1–11 (2016). https://doi.org/10.1007/s00006-015-0599-9
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DOI: https://doi.org/10.1007/s00006-015-0599-9