We propose a concept of a q-connection, where q is a Nth primitive root of unity, which is constructed by means of a graded q-differential algebra B with N-differential satisfying d N = 0. Having proved that the Nth power of a q-connection is the endomorphism of the left B-module F = β⊗Дд, where A is the subalgebra of elements of grading zero of β, and E is a left я-module, we give the definition of the curvature of a q-connection. We prove that the curvature satisfies the Bianchi identity. Assuming that E is a free finitely generated module we associate to a q-connection the matrix of this connection and the curvature matrix. We calculate the expression for the curvature matrix in terms of the entries of the matrix of q-connection. We also find the form of the Bianchi identity in terms of the curvature matrix and the matrix of a q-connection.
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References
V. Abramov, R. Kerner and B. Le Roy, Hypersymmetry: A ℤscript>3-generalization of supersymme-try, J. Math. Phys., 38(3), 1650–1669 (1997)
V. Abramov and R. Kerner, Exterior differentials of higher order and their covariant generalization, J. Math. Phys., 41(8), 5598–5614 (2000)
V. Abramov, On a graded q-differential algebra, J. Nonlinear Math. Phys., 13(Suppl.), 1–8 (2006)
V. Abramov, Generalization of superconnection in non-commutative geometry, Proc. Estonian Acad. Sci. Phys. Math., 55(1), 3–15 (2006)
C. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Interscience, New York, 1962
M. Dubois-Violette and R. Kerner, Universal q-differential calculus and q-analog of homolog-ical algebra, Acta Math. Univ. Comenian., 65, 175–188 (1996)
Dubois-Violette M. Lectures on graded differential algebras and noncommutative geometry, In Noncommutative differential geometry and its applications to physics: Proceedings of the workshop (Maeda, Yoshiaki., eds.). Kluwer, Dordrecht. Math. Phys. Stud., 23, 245–306 (2001)
M. Dubois-Violette, Lectures on differentials, generalized differentials and on some examples related to theoretical physics, math.QA/0005256.w
M. Kapranov, On the q-analog of homological algebra, math.QA/9611005
R. Kerner and V. Abramov, On certain realizations of q-deformed exterior differential calculus, Rep. Math. Phys., 43(1–2), 179–194 (1999)
V. Mathai and D. Quillen, Superconnections, Thom classes and equivariant differential forms, Topology, 25, 85–110 (1986)
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Abramov, V. (2009). Graded q-Differential Algebra Approach to q-Connection. In: Silvestrov, S., Paal, E., Abramov, V., Stolin, A. (eds) Generalized Lie Theory in Mathematics, Physics and Beyond. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85332-9_6
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