Character algebras of decorated $\operatorname{SL}_2(C)$–local systems

Abstract

Let $\mathcal{S}$ be a connected and locally 1–connected space, and let $\mathcal{ℳ}\subset \mathcal{S}$. A decorated ${SL}_2\left(ℂ\right)$–local system is an ${SL}_2\left(ℂ\right)$–local system on $\mathcal{S}$, together with a chosen element of the stalk at each component of $\mathcal{ℳ}$.

We study the decorated ${SL}_2\left(ℂ\right)$–character algebra of $\left(\mathcal{S},\mathcal{ℳ}\right)$: the algebra of polynomial invariants of decorated ${SL}_2\left(ℂ\right)$–local systems on $\left(\mathcal{S},\mathcal{ℳ}\right)$. The character algebra is presented explicitly. The character algebra is shown to correspond to the $ℂ$–algebra spanned by collections of oriented curves in $\mathcal{S}$ modulo local topological rules.

As an intermediate step, we obtain an invariant-theory result of independent interest: a presentation of the algebra of ${SL}_2\left(ℂ\right)$–invariant functions on $End{\left(\mathbb{V}\right)}^m\oplus {\mathbb{V}}^n$, where $\mathbb{V}$ is the tautological representation of ${SL}_2\left(ℂ\right)$.