## Algebraic & Geometric Topology

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

#### Abstract

Let $S$ be a connected and locally 1–connected space, and let $ℳ⊂S$. A decorated $SL2(ℂ)$–local system is an $SL2(ℂ)$–local system on $S$, together with a chosen element of the stalk at each component of $ℳ$.

We study the decorated $SL2(ℂ)$character algebra of $(S,ℳ)$: the algebra of polynomial invariants of decorated $SL2(ℂ)$–local systems on $(S,ℳ)$. The character algebra is presented explicitly. The character algebra is shown to correspond to the $ℂ$–algebra spanned by collections of oriented curves in $S$ modulo local topological rules.

As an intermediate step, we obtain an invariant-theory result of independent interest: a presentation of the algebra of $SL2(ℂ)$–invariant functions on $End(V)m⊕Vn$, where $V$ is the tautological representation of $SL2(ℂ)$.

#### Article information

Source
Algebr. Geom. Topol., Volume 13, Number 4 (2013), 2429-2469.

Dates
Received: 10 November 2011
Revised: 24 February 2013
Accepted: 11 March 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715646

Digital Object Identifier
doi:10.2140/agt.2013.13.2429

Mathematical Reviews number (MathSciNet)
MR3073924

Zentralblatt MATH identifier
06185357

#### Citation

Muller, Greg; Samuelson, Peter. Character algebras of decorated $\operatorname{SL}_2(C)$–local systems. Algebr. Geom. Topol. 13 (2013), no. 4, 2429--2469. doi:10.2140/agt.2013.13.2429. https://projecteuclid.org/euclid.agt/1513715646

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