# 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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topThomas 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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