Phorgy Phynance

Network Theory and Discrete Calculus – Edge Algebra

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This post is part of a series

In my last post, I noted that in following John Baez’ series, I’m finding the need to introduce operators that I haven’t previously used in any applications. In this post, I will introduce another. It turns out that we could get away without introducing this concept, but I think it helps motivate some things I will talk about later.

In all previous applications, the important algebra was a noncommutative graded differential algebra. The grading means that the degree of elements add when you multiply them together. For example, the product of two nodes (degree 0) is a node (degree 0+0), the product of a node (degree 0) and a directed edge (degree 1) is a directed edge (degree 0+1), and the product of a directed edge (degree 1) with another directed edge is a directed surface (degree 1+1).

Note the algebra of nodes is a commutative sub-algebra of the full noncommutative graded algebra.

There is another related commutative edge algebra with corresponding edge product.

The edge product is similar to the product of nodes in that it is a projection given by

$\mathbf{e}_\epsilon^{i,j} \circ \mathbf{e}_{\epsilon'}^{k,l} = \delta_{\epsilon,\epsilon'} \delta_{i,k} \delta_{j,l} \mathbf{e}_\epsilon^{i,j}.$

It is a projection because for an arbitrary discrete 1-form

\begin{aligned}\alpha = \sum_{i,j} \sum_{\epsilon\in [i,j]} \alpha_{i,j}^{\epsilon} \mathbf{e}_\epsilon^{i,j},\end{aligned}

we have

$\mathbf{e}_\epsilon^{i,j} \circ \alpha = \alpha_{i,j}^{\epsilon} \mathbf{e}_\epsilon^{i,j}$

and

$\mathbf{e}_\epsilon^{i,j} \circ \mathbf{e}_\epsilon^{i,j} = \mathbf{e}_\epsilon^{i,j}.$

The product of two discrete 1-forms is

\begin{aligned}\alpha\circ\beta = \sum_{i,j} \sum_{\epsilon\in [i,j]} \alpha_{i,j}^{\epsilon} \beta_{i,j}^{\epsilon} \mathbf{e}_\epsilon^{i,j}\end{aligned}.

I have not yet come across an application where the full edge algebra is needed. When the product does arise, one of the discrete 1-forms is usual the coboundary of a discrete 0-form, i.e.

$\alpha\circ d\phi.$

When this is the case, the edge product can be expressed as a (graded) commutator in the noncommutative graded algebra, i.e.

$\alpha\circ d\phi = [\alpha,\phi].$

An example of this will be seen when we examine electrical circuits.