Divergence zero quaternionic vector fields and Hamming graphs

Jasna Prezelj,Fabio Vlacci

doi:10.26493/1855-3974.2033.974

Abstract

We give a possible extension of the definition of quaternionic power series, partial derivatives and vector fields in the case of two (and then several) non commutative (quaternionic) variables. In this setting we also investigate the problem of describing zero functions which are not null functions in the formal sense. A connection between an analytic condition and a graph theoretic property of a subgraph of a Hamming graph is shown, namely the condition that polynomial vector field has formal divergence zero is equivalent to connectedness of subgraphs of Hamming graphs H ( d , 2) . We prove that monomials in variables z and w are always linearly independent as functions only in bidegrees ( p , 0), ( p , 1), (0, q ), (1, q ) and (2, 2).

Highlights

We give a possible extension of the definition of quaternionic power series, partial derivatives and vector fields in the case of two non commutative variables
Complex holomorphic vector fields with divergence zero represent an important tool for the description of the groups of volume preserving automorphisms of Cn with n > 1
In this paper we investigate generalizations of complex holomorphic vector fields in the quaternionic setting, and for this purpose we restrict our research to mappings represented by convergent quaternionic power series

Summary

Introduction

Complex holomorphic vector fields with divergence zero represent an important tool for the description of the groups of volume preserving automorphisms of Cn with n > 1 (we refer the reader to [1] and [2] for a thorough description of this topic). In this paper we investigate generalizations of complex holomorphic vector fields in the quaternionic setting, and for this purpose we restrict our research to mappings represented by convergent quaternionic power series. We introduce an alternative definition of partial derivative, namely as a first order approximation (which is not linear) and using this new notion of partial derivatives we define the corresponding divergence in the quaternionic setting. We show that quaternionic vector fields with divergence zero are bidegree full (see Section 2.2 for definition) and that the divergence zero condition on quaternionic vector fields is equivalent to finding connected subgraphs of Hamming graphs.

Convergent quaternionic power series

Basics on Hamming graphs

Quaternionic vector fields