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On webs in quantum type C

Part of: Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  26 February 2021

David E. V. Rose  and
Logan C. Tatham
David E. V. Rose*
Affiliation:
Department of Mathematics, The University of North Carolina at Chapel Hill, Phillips Hall CB #3250, UNC-CH, Chapel Hill, NC27599-3250, USAltatham@live.unc.edu
Logan C. Tatham
Affiliation:
Department of Mathematics, The University of North Carolina at Chapel Hill, Phillips Hall CB #3250, UNC-CH, Chapel Hill, NC27599-3250, USAltatham@live.unc.edu
*
e-mail: davidrose@unc.edu
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Abstract

We study webs in quantum type C, focusing on the rank three case. We define a linear pivotal category $\textbf {Web}(\mathfrak {sp}_{6})$ diagrammatically by generators and relations, and conjecture that it is equivalent to the category $\textbf {FundRep}(U_q(\mathfrak {sp}_{6}))$ of quantum $\mathfrak {sp}_{6}$ representations generated by the fundamental representations, for generic values of the parameter q. We prove a number of results in support of this conjecture, most notably that there is a full, essentially surjective functor $\textbf {Web}(\mathfrak {sp}_{6}) \rightarrow \textbf {FundRep}(U_q(\mathfrak {sp}_{6}))$ , that all $\textrm {Hom}$ -spaces in $\textbf {Web}(\mathfrak {sp}_{6})$ are finite-dimensional, and that the endomorphism algebra of the monoidal unit in $\textbf {Web}(\mathfrak {sp}_{6})$ is one-dimensional. The latter corresponds to the statement that all closed webs can be evaluated to scalars using local relations; as such, we obtain a new approach to the quantum $\mathfrak {sp}_{6}$ link invariants, akin to the Kauffman bracket description of the Jones polynomial.

Keywords

Websquantum groupspivotal category

MSC classification

Primary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 3 , June 2022 , pp. 793 - 832
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

This work was supported in part by Simons Collaboration Grant 523992: Research on knot invariants, representation theory, and categorification.

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