What would happen if we replaced the current monetary system, which is based on an abelian group [*] by a non-abelian currency system?
[*] If someone gives you x dollars, then y dollars, the result is the same as if you were given y dollars first, then x.
I’ve been puzzling about this for a few years. It raises lots of big questions. How would markets function differently? Might this lead to more efficient allocation of resources, at least in some instances? (At the very least, it’d completely change our notion of what it means to wealthy!) Might new forms of co-operation emerge? How would results in game theory change if we could use non-abelian payoffs?
More generally, it seems like this sort of idea might be used to look at all of economics through an interesting lens.
A nice toy model in this vein is to work with the group of 2 by 2 invertible matrices, with the group operation being matrix multiplication. By taking matrix logarithms, it can be shown that this model is a generalization of the current monetary system.
Electronic implementation of non-abelian money would be a snap. The social implementation might be a bit tougher, however – convincing people that their net wealth should be a matrix would be a tough sell, at least initially. Still, if non-abelian money changed some key results from economics, then in some niches it may be advantageous to make the switch, and possible to convince people that this is a good idea.
(It should, of course, be noted that there are in practice already many effects which make money act in a somewhat non-abelian fashion, e.g., inflation. From the point of view of this post, these are kludges: I’m talking about changing the underlying abstraction to a new one.)