Character algebras of decorated SL2(C)–local systems

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

We study the decorated ${SL}_{2} (ℂ)$ –character algebra of $(S, ℳ)$ : the algebra of polynomial invariants of decorated ${SL}_{2} (ℂ)$ –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 ${SL}_{2} (ℂ)$ –invariant functions on $End {(V)}^{m} \oplus V^{n}$ , where $V$ is the tautological representation of ${SL}_{2} (ℂ)$ .

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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

References

Publication

Received: 10 November 2011

Revised: 24 February 2013

Accepted: 11 March 2013

Published: 4 July 2013

Authors

Greg Muller
Mathematics Department Louisiana State University 7250 Perkins Rd Apt 217 Baton Rouge, LA 70808 USA
https://www.math.lsu.edu/~gmuller/
Peter Samuelson
Department of Mathematics University of Toronto Bahen Centre 40 George St. Rm 6290 Toronto, ON M5S 2E4 Canada
http://www.math.toronto.edu/cms/samuelson-peter/

Volume 13, issue 4 (2013)

Greg Muller and Peter Samuelson

Abstract