In the mathematics of moduli theory, given an algebraic, reductive, Lie group
Contents
More precisely,
Formulation
Formally, and when the algebraic group is defined over the complex numbers
Here more generally one can consider algebraically closed fields of prime characteristic. In this generality, character varieties are only algebraic sets and are not actual varieties. To avoid technical issues, one often considers the associated reduced space by dividing by the radical of 0 (eliminating nilpotents). However, this does not necessarily yield an irreducible space either. Moreover, if we replace the complex group by a real group we may not even get an algebraic set. In particular, a maximal compact subgroup generally gives a semi-algebraic set. On the other hand, whenever
Examples
For example, if
Another example, also studied by Vogt and Fricke-Klein is the case with
Variants
This is not necessarily the same construction as the Culler-Shalen character variety (generated by evaluations of traces), although when the
on
even if
For instance, for a free group of rank 2 and
But the trace algebra is a strictly small subalgebra (there are less invariants). This provides an involutive action on the torus that needs to be accounted for to yield the Culler-Shalen character variety. The involution on this torus yields a 2-sphere. The point is that up to
Connection to geometry
There is an interplay between these moduli and the moduli of principal bundles, vector bundles, Higgs bundles, and geometric structures on topological spaces, given generally by the observation that, at least locally, equivalent objects in these categories are parameterized by conjugacy classes of holonomy homomorphisms. In other words, with respect to a base space
Connection to skein modules
The coordinate ring of the character variety has been related to skein modules in knot theory. The skein module is roughly a deformation (or quantization) of the character variety. It is closely related to topological quantum field theory in dimension 2+1.