On the q-exponential of matrix q-Lie algebras

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 tau, 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 x n matrix, a so-called q-Lie group, or manifold, usually with q-determinant 1. The corresponding matrix multiplication is twisted under tau, 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 su(q)(4), and its corresponding q-Lie group. We conclude with an example of system of equations with Ward number coeficients