Representations of U(1,q) and Constructive Quaternion Tensor Products

TL;DR

This paper explores the representation theory of U(1,q), highlighting its implications for quaternionic physics models, and introduces a quaternion tensor product with new group representations.

Contribution

It provides a detailed analysis of U(1,q) representations and defines a quaternion tensor product, advancing quaternionic group theory and its applications in physics.

Findings

Eigenvalues are right-eigenvalues in U(1,q) representations

Only complex scalar products are consistent

A new quaternion tensor product yields additional spin representations

Abstract

The representation theory of the group U(1,q) is discussed in detail because of its possible application in a quaternion version of the Salam-Weinberg theory. As a consequence, from purely group theoretical arguments we demonstrate that the eigenvalues must be right-eigenvalues and that the only consistent scalar products are the complex ones. We also define an explicit quaternion tensor product which leads to a set of additional group representations for integer ``spin''.