Poisson Algebra of Wilson Loops and Derivations of Free Algebras

Abstract

We describe a finite analogue of the Poisson algebra of Wilson loops in Yang-Mills theory. It is shown that this algebra arises in an apparently completely different context; as a Lie algebra of vector fields on a non-commutative space. This suggests that non-commutative geometry plays a fundamental role in the manifestly gauge invariant formulation of Yang-Mills theory. We also construct the deformation of the loop algebra induced by quantization, in the large N_c limit.