Character algebras of decorated SL_2(C)-local systems

TL;DR

This paper explicitly describes the algebra of functions on decorated SL_2(C)-local systems on a CW-complex, linking it to collections of curves and providing an invariant-theory result for SL_2(C).

Contribution

It provides a new explicit presentation of the decorated character algebra and relates it to graphical curve collections, advancing understanding of local system invariants.

Findings

The character algebra is generated by collections of oriented curves in the surface.

An explicit presentation of the algebra of SL_2(C)-invariant functions on End(V)^m + V^n.

The algebra of decorated local systems is isomorphic to an algebra spanned by graphical curve collections.

Abstract

Let S be a path-connected, locally-compact CW-complex, and let M be a subcomplex with finitely-many components. A `decorated SL_2(C)-local system' is an SL_2(C)-local system on S, together with a choice of `decoration' at each component of M (a section of the stalk of an associated vector bundle). We study the (decorated SL_2(C)-)character algebra of (S,M), those functions on the space of decorated SL_2(C)-local systems on (S,M) which are regular with respect to the monodromy. The character algebra is presented explicitly. The character algebra is then shown to correspond to the algebra spanned by collections of oriented curves in S modulo simple graphical 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 + V^n, where V is the tautological representation of SL_2(C).