Division algebra valued energized simplicial complexes

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.