Division algebra valued energized simplicial complexes

TL;DR

This paper explores connection Laplacians valued in division algebras, analyzing their determinants, spectra, and geometric properties when defined on simplicial complexes with values in various algebraic structures.

Contribution

It introduces a framework for connection Laplacians over division algebras, generalizing previous scalar cases and providing new insights into their spectral and topological properties.

Findings

Dieudonne determinants equal to abelianization of product of field values

Spectrum of L(G,h) studied over complex numbers with geometric analysis

Connected components form non-compact Kaehler manifolds with computable fundamental groups

Abstract

We look at connection Laplacians L,g defined by a field h:G to K, where G is a finite set of sets and K is a normed division ring which does not need to be commutative, nor associative but has a conjugation leading to the norm as the square root of h^* h. The target space K can be a normed real division algebra like the quaternions or an algebraic number field like a quadratic field. For parts of the results we can even assume K to be a Banach algebra like an operator algebra on a Hilbert space. The K-valued function h on G then defines connection matrices L,g in which the entries are in K. We show that the Dieudonne determinants of L and g are both equal to the abelianization of the product of all the field values on G. If G is a simplicial complex and h takes values in the units U of K, then g^* is the inverse of L and the sum of the energy values is equal to the sum of the Green function entries g(x,y). If K is the field C of complex numbers, we can study the spectrum of L(G,h) in dependence of the field h. The set of matrices with simple spectrum defines a |G|-dimensional non-compact Kaehler manifold that is disconnected in general and for which we can compute the fundamental group of each connected component.