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Graded q-Differential Algebra Approach to q-Connection

In: Generalized Lie Theory in Mathematics, Physics and Beyond

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  • Viktor Abramov

    (University of Tartu, Liivi 2, Institute of Mathematics)

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.

Suggested Citation

  • Viktor Abramov, 2009. "Graded q-Differential Algebra Approach to q-Connection," Springer Books, in: Sergei Silvestrov & Eugen Paal & Viktor Abramov & Alexander Stolin (ed.), Generalized Lie Theory in Mathematics, Physics and Beyond, chapter 6, pages 71-79, Springer.
    • Handle: RePEc:spr:sprchp:978-3-540-85332-9_6
    DOI: 10.1007/978-3-540-85332-9_6
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