Erlangen Program at Large-2.5: Induced Representations and Hypercomplex Numbers

TL;DR

This paper explores the construction of induced representations of SL(2,R) using hypercomplex numbers, revealing new connections between group actions, hypercomplex algebra, and unitary representations in mathematical analysis.

Contribution

It introduces a novel framework linking hypercomplex numbers with induced representations of SL(2,R), expanding the understanding of hypercomplex analytic functions and their group-theoretic properties.

Findings

Hypercomplex numbers correspond to group actions on the half-plane.

Induced representations on hypercomplex-valued function spaces are constructed.

Raising and lowering operators involve hypercomplex coefficients.

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