On the q-exponential of matrix q-Lie algebras

Special Matrices (2017)

• Volume: 5, Issue: 1, page 36-50
• ISSN: 2300-7451

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Abstract

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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.

How to cite

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Thomas Ernst. "On the q-exponential of matrix q-Lie algebras." Special Matrices 5.1 (2017): 36-50. <http://eudml.org/doc/288019>.

@article{ThomasErnst2017,
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.},
author = {Thomas Ernst},
journal = {Special Matrices},
keywords = {Ring morphism; q-determinant; Nova q-addition; q-exponential function; q-Lie algebra; q-trace; biring; ring morphism; $q$-determinant; nova $q$-addition; -exponential function; -Lie algebra; -trace},
language = {eng},
number = {1},
pages = {36-50},
title = {On the q-exponential of matrix q-Lie algebras},
url = {http://eudml.org/doc/288019},
volume = {5},
year = {2017},
}

TY - JOUR
AU - Thomas Ernst
TI - On the q-exponential of matrix q-Lie algebras
JO - Special Matrices
PY - 2017
VL - 5
IS - 1
SP - 36
EP - 50
AB - 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.
LA - eng
KW - Ring morphism; q-determinant; Nova q-addition; q-exponential function; q-Lie algebra; q-trace; biring; ring morphism; $q$-determinant; nova $q$-addition; -exponential function; -Lie algebra; -trace
UR - http://eudml.org/doc/288019
ER -

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