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.
@article{bwmeta1.element.doi-10_1515_spma-2017-0003,
author = {Thomas Ernst},
title = {On the q-exponential of matrix q-Lie algebras},
journal = {Special Matrices},
volume = {5},
year = {2017},
pages = {36-50},
zbl = {06694928},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_spma-2017-0003}
}
Thomas Ernst. On the q-exponential of matrix q-Lie algebras. Special Matrices, Tome 5 (2017) pp. 36-50. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_spma-2017-0003/