There is an algebra defined on a two dimensional manifold, known as the Skein algebra, which has as elements the simple closed curves of the manifold. Just like with numbers, there's a way to add, subtract and multiply elements. Unfortunately division is not allowed in the Skein algebra, which is why we introduced the notion of the Localized Skein Algebra, where we define a way to invert elements so that dividing is possible. These algebras have infinitely many elements, may not be commutative and in fact may have torsion, which makes them a hard object to study.
This work is mainly centered in reducing these algebras to something more manageable. We have shown that for any space, its Localized Skein Algebra is a Frobenius extension of its Localized Character Ring, which means that any element of the algebra can be rewritten as a finite linear combination of a finite subset of basis elements, multiplied by elements that do commute. The importance of this result is that it solves this problem of noncommutativity, by rewriting anything that doesn't commute, as elements of a small set which can be controlled, along with elements that commute and behave nicely, making the Skein algebra far more manageable.