In this paper we give a topological interpretation and diagrammatic calculus for the rank (n − 2) Askey-Wilson algebra by proving there is an explicit isomorphism with the Kauffman bracket skein algebra of the (n + 1)-punctured sphere. To do this we consider the Askey-Wilson algebra in the braided tensor product of n copies of either the quantum group Uq(sl 2 ) or the reflection equation algebra. We then use the isomorpism of the Kauffman bracket skein algebra of the (n + 1)-punctured sphere with the Uq(sl 2 ) invariants of the Aleeksev moduli algebra to complete the correspondence. We also find the graded vector space dimension of the Uq(sl 2 ) invariants of the Aleeksev moduli algebra and apply this to finding a presentation of the skein algebra of the five-punctured sphere and hence also find a presentation for the rank 2 Askey-Wilson algebra. Contents 1. Introduction 1 2. Askey-Wilson Algebras 5 3. Braided Tensor Product of copies of the locally finite part of U q (sl 2 ) 8 4. Moduli algebras 14 5. Skein Algebras 17 6. Commutator Relations 20 7. Action of the braid group 21 8. Graded Dimensions 23 9. Presentation of the Skein Algebra of the Five-Punctured Sphere 25 A. Appendix 34 References 37
We prove that the skein categories of Walker-Johnson-Freyd satisfy excision. This allows us to conclude that skein categories are k-linear factorisation homology and taking the free cocompletion of skein categories recovers locally finitely presentable factorisation homology. An application of this is that the skein algebra of a punctured surface related to any quantum group with generic parameter gives a quantisation of the associated character variety. † The Kauffman bracket skein algebra is the case G = SL 2 .
We compute the factorization homology of the four-punctured sphere and punctured torus over the quantum group [Formula: see text] explicitly as categories of equivariant modules using the framework developed by Ben-Zvi et al. We identify the algebra of [Formula: see text]-invariants (quantum global sections) with the spherical double affine Hecke algebra of type [Formula: see text], in the four-punctured sphere case, and with the “cyclic deformation” of [Formula: see text] in the punctured torus case. In both cases, we give an identification with the corresponding quantum Teichmüller space as proposed by Teschner and Vartanov as a quantization of the moduli space of flat connections.
We study the skein algebra of the genus 2 surface and its action on the skein module of the genus 2 handlebody. We compute this action explicitly, and we describe how the module decomposes over certain subalgebras in terms of polynomial representations of double affine Hecke algebras. Finally, we show that this algebra is isomorphic to the t = q specialisation of the genus two spherical double affine Hecke algebra recently defined by Arthamonov and Shakirov.
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