Algebraic & Geometric Topology

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

Greg Muller and Peter Samuelson

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Abstract

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

We study the decorated SL2()character algebra of (S,): the algebra of polynomial invariants of decorated SL2()–local systems on (S,). The character algebra is presented explicitly. The character algebra is shown to correspond to the –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()–invariant functions on End(V)mVn, where V is the tautological representation of SL2().

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

Subjects
Primary: 13A50: Actions of groups on commutative rings; invariant theory [See also 14L24] 14D20: Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13} 57M27: Invariants of knots and 3-manifolds 57M07: Topological methods in group theory

Keywords
local systems rings of invariants mixed invariants mixed concomitants skein algebra cluster algebra quantum cluster algebra quantum torus triangulation of surfaces

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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