Character algebras of decorated SL2(C)–local systems

Muller, Greg¹ ; Samuelson, Peter²

¹Mathematics Department, Louisiana State University, 7250 Perkins Rd, Apt 217, Baton Rouge, LA 70808, USA
²Department of Mathematics, University of Toronto, Bahen Centre, 40 George St. Rm 6290, Toronto, ON M5S 2E4, Canada

Algebraic and Geometric Topology, Tome 13 (2013) no. 4, pp. 2429-2469

Résumé

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).

Détail

DOI : 10.2140/agt.2013.13.2429

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

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