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Special Matrices

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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


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


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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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