#### Volume 13, issue 4 (2013)

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

### Greg Muller and Peter Samuelson

Algebraic & Geometric Topology 13 (2013) 2429–2469
##### Abstract

Let $\mathsc{S}$ be a connected and locally 1–connected space, and let $\mathsc{ℳ}\subset \mathsc{S}$. A decorated ${SL}_{2}\left(ℂ\right)$–local system is an ${SL}_{2}\left(ℂ\right)$–local system on $\mathsc{S}$, together with a chosen element of the stalk at each component of $\mathsc{ℳ}$.

We study the decorated ${SL}_{2}\left(ℂ\right)$character algebra of $\left(\mathsc{S},\mathsc{ℳ}\right)$: the algebra of polynomial invariants of decorated ${SL}_{2}\left(ℂ\right)$–local systems on $\left(\mathsc{S},\mathsc{ℳ}\right)$. The character algebra is presented explicitly. The character algebra is shown to correspond to the $ℂ$–algebra spanned by collections of oriented curves in $\mathsc{S}$ 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 ${SL}_{2}\left(ℂ\right)$.

##### Keywords
local systems, rings of invariants, mixed invariants, mixed concomitants, skein algebra, cluster algebra, quantum cluster algebra, quantum torus, triangulation of surfaces
##### Mathematical Subject Classification 2010
Primary: 13A50, 14D20, 57M27, 57M07