Graded q-Differential Algebra Approach to q-Connection

Viktor Abramov ()
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Viktor Abramov: University of Tartu, Liivi 2, Institute of Mathematics

Chapter 6 in Generalized Lie Theory in Mathematics, Physics and Beyond, 2009, pp 71-79 from Springer

Abstract: 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.

Keywords: Vector Bundle; Differential Form; Bianchi Identity; Homogeneous Element; Differential Algebra (search for similar items in EconPapers)
Date: 2009
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-540-85332-9_6

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DOI: 10.1007/978-3-540-85332-9_6

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