On the q-exponential of matrix q-Lie algebras

Thomas Ernst 1
  • 1 Department of Mathematics, Uppsala University, P.O. Box 480, SE-751 06 , Uppsala, Sweden


In this paper, we define several new concepts in the borderline between linear algebra, Lie groups and q-calculus.We first introduce the ring epimorphism r, the set of all inversions of the basis q, and then the important q-determinant and corresponding q-scalar products from an earlier paper. Then we discuss matrix q-Lie algebras with a modified q-addition, and compute the matrix q-exponential to form the corresponding n × n matrix, a so-called q-Lie group, or manifold, usually with q-determinant 1. The corresponding matrix multiplication is twisted under τ, which makes it possible to draw diagrams similar to Lie group theory for the q-exponential, or the so-called q-morphism. There is no definition of letter multiplication in a general alphabet, but in this article we introduce new q-number systems, the biring of q-integers, and the extended q-rational numbers. Furthermore, we provide examples of matrices in suq(4), and its corresponding q-Lie group. We conclude with an example of system of equations with Ward number coeficients.

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  • [2] T. Ernst, q-deformed matrix pseudo-groups. Royal Flemish Academy of Belgium (2010), 151-162

  • [3] T. Ernst, A comprehensive treatment of q-calculus, Birkhäuser 2012.

  • [4] T. Ernst, An umbral approach to find q-analogues of matrix formulas, Linear Algebra Appl. 439 (2013), 1167-1182.

  • [5] T. Ernst, Multiplication formulas for q-Appell polynomials and the multiple q-power sums. Ann. Univ. Marie Curie (2016)

  • [6] W. Pfeifer, The Lie algebras su(N). An introduction. Birkhäuser (2003)

  • [7] J.D.Talman, Special functions. A group theoretic approach. The Mathematical Physics Monograph Series. New York- Amsterdam: W.A. Benjamin, 1968.


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