Heisenberg groups and their automorphisms over algebras with central involution

TL;DR

This paper constructs and analyzes Heisenberg groups over algebras with central involution, exploring their automorphisms, representations, and applications to finite and real quadratic spaces, using examples like quaternion group algebra.

Contribution

It introduces a new framework for Heisenberg groups over algebras with central involution and develops models for their automorphisms and representations, including a pseudo-differential operator approach.

Findings

Finite models for quadratic spaces of dimension 4 or less

Construction of automorphism groups for these Heisenberg groups

A pseudo-differential operator for unified treatment over finite and real fields

Abstract

Heisenberg groups over algebras with central involution and their automorphism groups are constructed. The complex quaternion group algebra over a prime field is used as an example. Its subspaces provide finite models for each of the real and complex quadratic spaces with dimension 4 or less. A model for the representations of these Heisenberg groups and automorphism groups is constructed. A pseudo-differential operator enables a parallel treatment of spaces defined over finite and real fields.