Holonomy and (stated) skein algebras in combinatorial quantization

Abstract

The algebra L g,n (H) was introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche and quantizes the character variety of the Riemann surface Σ g,n \ D (D is an open disk). In this article we define a holonomy map in that quantized setting, which associates a tensor with components in L g,n (H) to tangles in (Σ g,n \D) × [0, 1], generalizing previous works of Buffenoir-Roche and Bullock-Frohman-Kania-Bartoszynska. We show that holonomy behaves well for the stack product and the action of the mapping class group; then we specialize this notion to links in order to define a generalized Wilson loop map. Thanks to the holonomy map, we give a geometric interpretation of the vacuum representation of L g,0 (H) on L 0,g (H). Finally, the general results are applied to the case H = U q 2 (sl 2 ) in relation to skein theory and the most important consequence is that the stated skein algebra of a compact oriented surface with just one boundary edge is isomorphic to L g,n U q 2 (sl 2 ) . Throughout the paper we use a graphical calculus for tensors with coefficients in L g,n (H) which makes the computations and definitions very intuitive.

Relating stated skein algebras and internal skein algebras

“…When V = O q 2 (SL 2 )-comod f in , we show that these two algebras are isomorphic. This result was already stated in [LY20,Theorem 4.4] and [GJS19, Remark 2.21], and both algebras are known to be isomorphic to the Alekseev-Grosse-Schomerus-algebras, see [Fai20,Theorem 5.3] and [Kor20] for the stated skein algebra side and [BBJ18] for the internal skein algebra side. The author has not been able to find an explicit and natural relation though, in particular keeping track of the product inversion coming from seeing the boundary edge at the right.…”

Unrestricted quantum moduli algebras, II: Noetherianity and simple fraction rings at roots of $1$

“…where ℓ is the pivotal element. Then Theorem 2.14 (1) and (41) imply that U ǫ is a free Z 0 (U lf ǫ )[ℓ l ]-module of rank 2 m l dimg , and Theorem 2.2 (2) says that it is also the direct sum of 2 m copies of the (free…”

“…Besides, the quantum moduli algebras are very interesting objects in themselves. They are now recognized as central objects from the viewpoints of factorization homology [18], (stated) skein theory [19,41,29] and, as already said, the mapping class group representations associated to topological quantum field theories [40].…”

Stated skein algebras and their representations

“…Theorem 2.31. (Lê [Le18] for the bigon and the triangle, Faitg [Fai20] for Σ 0 g,n , K. [Kor20, Theorem 1.1] in general) For a connected marked surface Σ with non trivial marking and for a finite presentation (G, RL) of Π 1 (Σ, V), the stated skein algebra S A (Σ) has finite presentation with generators the elements β ij with β ∈ G and i, j = ±, and relations the trivial loops relations (13), the q-determinant relations (14) and the arc exchange relations.…”