Green functions of Energized complexes
Abstract
If h is a ringvalued function on a simplicial complex G we can define two matrices L and g, where the matrix entries are the h energy of homoclinic intersections. We know that the sum over all h values on G is equal to the sum of the Green matrix entries g(x,y). We also have already seen that that the determinants of L or g are both the product of the h(x). In the case where h(x) is the parity of dimension, the sum of the energy values was the standard Euler characteristic and the determinant was a unit. If h(x) was the unit in the ring then L,g are integral quadratic forms which are isospectral and inverse matrices of each other. We prove here that the quadratic energy expression summing over all pairs h(x)^* h(y) of intersecting sets is a signed sum of squares of Green function entries. The quadratic energy expression is Wu characteristic in the case when h is dimension parity. For general h, the quadratic energy expression resembles an Ising Heisenberg type interaction. The conjugate of g is the inverse of L if h takes unit values in a normed ring or in the group of unitary operators in an operator algebra.
 Publication:

arXiv eprints
 Pub Date:
 October 2020
 arXiv:
 arXiv:2010.09152
 Bibcode:
 2020arXiv201009152K
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Discrete Mathematics;
 05C10;
 57M1
 EPrint:
 14 ages