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Special Matrices

Editor-in-Chief: da Fonseca, Carlos Martins

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CiteScore 2017: 0.29

SCImago Journal Rank (SJR) 2017: 0.295
Source Normalized Impact per Paper (SNIP) 2017: 0.481

Mathematical Citation Quotient (MCQ) 2017: 0.17

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2300-7451
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On the q-exponential of matrix q-Lie algebras

Thomas Ernst
Published Online: 2017-01-20 | DOI: https://doi.org/10.1515/spma-2017-0003

Abstract

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.

Keywords: Ring morphism; q-determinant; Nova q-addition; q-exponential function; q-Lie algebra; q-trace; biring

MSC 2010: Primary 17B99; Secondary 17B37; 33D15

References

  • [1] C.W.Conatser, Contractions of the lowdimensional real Lie algebras. J. Math. Physics 13, (1972), 196-203Google Scholar

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

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

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

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

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

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

About the article

Received: 2016-07-04

Accepted: 2016-09-27

Published Online: 2017-01-20

Published in Print: 2017-01-26


Citation Information: Special Matrices, Volume 5, Issue 1, Pages 36–50, ISSN (Online) 2300-7451, DOI: https://doi.org/10.1515/spma-2017-0003.

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© 2017. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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