[Paper Review] On analytic analogues of quantum groups

Craig Smith|arXiv (Cornell University)|Jun 27, 2018

Algebraic structures and combinatorial models11 references3 citations

TL;DR

This paper introduces a new construction of analytic quantum groups over non-Archimedean and Archimedean fields using analytic Nichols algebras and Majid's double-bosonisation, establishing braided monoidal categories of representations. It proves rigidity results via bounded cohomology, laying groundwork for a p-adic Drinfel'd-Kohno Theorem and suggesting new braid group representations involving p-adic special functions.

ABSTRACT

In this paper we present a new construction of analytic analogues of quantum groups over non-Archimedean fields and construct braided monoidal categories of their representations. We do this by constructing analytic Nichols algebras and use Majid's double-bosonisation construction to glue them together. We then go on to study the rigidity of these analytic quantum groups as algebra deformations of completed enveloping algebras through bounded cohomology. This provides the first steps towards a $p$-adic Drinfel'd-Kohno Theorem, which should relate this work to Furusho's $p$-adic Drinfel'd associators. Finally, we adapt these constructions to working over Archimedean fields.

Motivation & Objective

• To construct analytic analogues of quantum groups over non-Archimedean fields for arbitrary Kac-Moody Lie algebras, overcoming limitations of prior constructions.
• To establish braided monoidal categories of representations for these analytic quantum groups, enabling new braid group representations.
• To study the rigidity of these quantum groups as algebra deformations of completed enveloping algebras using bounded cohomology.
• To lay the foundation for a p-adic Drinfel'd-Kohno Theorem by relating these structures to Furusho's p-adic Drinfel'd associators.
• To extend the construction to Archimedean fields and explore connections to special analytic functions like p-adic multiple polylogarithms.

Proposed method

• Construct analytic Nichols algebras over non-Archimedean fields as quotients of completed tensor algebras by universal Hopf ideals or the radical of a duality pairing.
• Apply Majid's double-bosonisation construction to glue analytic Nichols algebras of positive and negative parts into full analytic quantum groups.
• Use an alternate description as quotients of Drinfel'd doubles to obtain braided monoidal categories of representations isomorphic to a BGG-like category O.
• Employ bounded cohomology to prove rigidity theorems, showing that algebra deformations of completed enveloping algebras are trivial under a cohomological vanishing assumption.
• Adapt the constructions to Archimedean fields by defining analytic Nichols algebras and quantum groups in the IndBanach category over R or C.
• Utilize duality pairings and coactions to define categories of crossed modules and establish fully faithful functors into module categories of the analytic quantum groups.

Experimental results

Research questions

• RQ1 Can analytic quantum groups be systematically constructed over non-Archimedean fields for arbitrary Kac-Moody Lie algebras using analytic Nichols algebras?
• RQ2 Do these analytic quantum groups admit braided monoidal categories of representations that generalize the BGG category O?
• RQ3 Are these analytic quantum groups rigid as algebra deformations of completed enveloping algebras, and what cohomological conditions ensure this?
• RQ4 Can the construction be extended to Archimedean fields, and what new representation-theoretic structures emerge?
• RQ5 Is there a connection between these analytic quantum groups and p-adic Drinfel'd associators, potentially leading to a p-adic Drinfel'd-Kohno Theorem?

Key findings

• Analytic Nichols algebras are constructed in two equivalent ways: as quotients by a universal Hopf ideal and as quotients by the radical of a duality pairing, ensuring compatibility for double-bosonisation.
• The double-bosonisation construction yields IndBanach Hopf algebras called analytic quantum groups, which are completions of the positive and negative parts of quantum enveloping algebras.
• Although the standard R-matrix does not converge in the analytic quantum groups, a braided monoidal category of representations is constructed via an alternate quotient description of the Drinfel'd double.
• Rigidity theorems are proven using bounded cohomology: under a cohomological vanishing assumption, any algebra deformation of a completed enveloping algebra is trivial and unique up to conjugation.
• The construction extends to Archimedean fields, yielding analytic quantum groups and braided module categories, with the category OΨ shown to be braided via coaction and duality structures.
• The framework suggests new braid group representations and potential appearances of p-adic multiple polylogarithms and quantum dilogarithms in the braiding of modules.