Representations of the Kauffman bracket skein algebra I: invariants and miraculous cancellations

Abstract

We study finite-dimensional representations of the Kauffman bracket skein algebra of a surface S. In particular, we construct invariants of such irreducible representations when the underlying parameter q = e 2πi is a root of unity. The main one of these invariants is a point in the character variety consisting of group homomorphisms from the fundamental group π 1 (S) to SL 2 (C), or in a twisted version of this character variety. The proof relies on certain miraculous cancellations that occur for the quantum trace homomorphism constructed by the authors. These miraculous cancellations also play a fundamental role in subsequent work of the authors, where novel examples of representations of the skein algebra are constructed.For an oriented surface S of finite topological type and for a Lie group G, many areas of mathematics involve the character varietywhere G acts on homomorphisms by conjugation. For G = SL 2 (C), Turaev [Tu 1 ] showed that the corresponding character variety R SL 2 (C) (S) can be quantized by the Kauffman bracket skein algebra of the surface; see also [BuFK 1 , BuFK 2 , PrS]. In fact, if one follows the physical tradition that a quantization of a space X replaces the commutative algebra of functions on X by a non-commutative algebra of operators on a Hilbert space, the points of an actual quantization of the character variety R SL 2 (C) (S) should be representations of the Kauffman bracket skein algebra.This article studies finite-dimensional representations of the skein algebra of a surface. The Kauffman bracket skein algebra S A (S) depends on a parameter A = e πi ∈ C − {0}, and is defined as follows. One first considers the vector space freely generated by all isotopy classes of framed links in the thickened surface S × [0, 1], and then one takes the quotient of this space by two relations: the main one is the skein relation thatwhenever the three links K 1 , K 0 and K ∞ ⊂ S × [0, 1] differ only in a little ball where they are as represented on Figure 1; the second relation states that [O] = −(A 2 + A −2 )[∅] for the trivial framed knot O and the empty link ∅. The algebra multiplication is defined by superposition of skeins. See §2 for details.Our goal is to study representations of the skein algebra, namely algebra homomorphisms ρ : S A (S) → End(V ) where V is a finite-dimensional vector space over C. See [BoW 2 ] for an interpretation of such representations as generalizations of the Kauffman bracket Date: June 5, 2018.

Unicity for representations of the Kauffman bracket skein algebra

“…See Section 3.3 for details. The definition of T (α) involves Bonahon and Wong's threading map [3]. As the C-vector space K ζ (F ) has basis {T (α) | α ∈ S }, hence it is enough to compute the trace of each T (α).…”

Quantum Teichmüller Spaces and Quantum Trace Map

“…Remark 6.9. We need only a special case of [BW2,Proposition 29], namely, the case when α is ∆-simple, and no cabling is applied to α. Although the proof of [BW2,Proposition 29] has long calculations, this special case is much simpler and follows almost immediately from the definition of tr ∆ q in [BW1].…”

“…Remark 7.4. Again we need only a special case of [BW2,Proposition 29] when no cabling is applied to α. This special case is very simple and follows almost immediately from the definition of tr ∆ q in [BW1].…”

“…The introduction of the quantum trace map is a breakthrough in the study of skein algebras; it embeds the skein algebraS into quantum tori which have simple algebraic structure. For example, representations ofS are studied via the quantum trace map in [BW2]. We will use quantum trace maps to the study of skein modules of knot complements in future work.…”

On the skein module of the product of a surface and a circle

“…There has been a lot of work on the algebra structure on the skein module of a surface times an interval (sometimes with coefficients different from Q(A)), and also on the connection between skein modules with character varieties. We would like to mention [BW,FKL,Le2] for some recent papers on this.…”