Relating Stated Skein Algebras and Internal Skein Algebras

Symmetry Integrability and Geometry-Methods and Applications

article

Benjamin Haïoun¹ 

[1] Institut de Mathématiques de Toulouse

关键词: quantum invariants;    skein theory;    category theory.;   

DOI  :  10.3842/SIGMA.2022.042

来源: National Academy of Science of Ukraine

【 摘 要 】

We give an explicit correspondence between stated skein algebras, which are defined via explicit relations on stated tangles in [Costantino F., Lê T.T.Q., arXiv:1907.11400], and internal skein algebras, which are defined as internal endomorphism algebras in free cocompletions of skein categories in [Ben-Zvi D., Brochier A., Jordan D., J. Topol. 11 (2018), 874-917, arXiv:1501.04652] or in [Gunningham S., Jordan D., Safronov P., arXiv:1908.05233]. Stated skein algebras are defined on surfaces with multiple boundary edges and we generalise internal skein algebras in this context. Now, one needs to distinguish between left and right boundary edges, and we explain this phenomenon on stated skein algebras using a half-twist. We prove excision properties of multi-edges internal skein algebras using excision properties of skein categories, and agreeing with excision properties of stated skein algebras when $\mathcal{V} = \mathcal{U}_{q^2}(\mathfrak{sl}_2)\text{-}{\rm mod}^{\rm fin}$. Our proofs are mostly based on skein theory and we do not require the reader to be familiar with the formalism of higher categories.

【 授权许可】

Unknown