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ISSN (electronic): 1472-2739
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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

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)m Vn, where V is the tautological representation of SL2().

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
Received: 10 November 2011
Revised: 24 February 2013
Accepted: 11 March 2013
Published: 4 July 2013
Greg Muller
Mathematics Department
Louisiana State University
7250 Perkins Rd
Apt 217
Baton Rouge, LA 70808
Peter Samuelson
Department of Mathematics
University of Toronto
Bahen Centre
40 George St. Rm 6290
Toronto, ON M5S 2E4