Erlangen Program at Large-2.5: Induced Representations and Hypercomplex Numbers
Abstract
In the search for hypercomplex analytic functions on the half-plane, we review the construction of induced representations of the group G=SL(2,R). Firstly we note that G-action on the homogeneous space G/H, where H is any one-dimensional subgroup of SL(2,R), is a linear-fractional transformation on hypercomplex numbers. Thus we investigate various hypercomplex characters of subgroups H. The correspondence between the structure of the group SL(2,R) and hypercomplex numbers can be illustrated in many other situations as well. We give examples of induced representations of SL(2,R) on spaces of hypercomplex valued functions, which are unitary in some sense. Raising/lowering operators for various subgroup prompt hypercomplex coefficients as well. The paper contains both English and Russian versions. Keywords: induced representation, unitary representations, SL(2,R), semisimple Lie group, complex numbers, dual numbers, double numbers, Moebius transformations, split-complex numbers, parabolic numbers, hyperbolic numbers, raising/lowering operators, creation/annihilation operators
Cite
@article{arxiv.0909.4464,
title = {Erlangen Program at Large-2.5: Induced Representations and Hypercomplex Numbers},
author = {Vladimir V. Kisil},
journal= {arXiv preprint arXiv:0909.4464},
year = {2015}
}
Comments
LaTeX2e; 17 pp + 13 pp of a source code; 5 EPS pictures in two Figures; v2: minor improvements and corrections; v3: a section on raising/lowering operators is added; v4: typos are fixed; v5: Introduction is added, open problems are expanded.v6: Russian translation is added, references areupdated, NoWeb and C++ source codes are added as ancillary files. arXiv admin note: substantial text overlap with arXiv:0707.4024