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Abstract Our work is motivated by obtaining solutions to the quantum reflection equation (qRE) by categorical methods. To start, given a braided monoidal category$${\mathcal {C}}$$and$${\mathcal {C}}$$-module category$${\mathcal {M}}$$, we introduce a version of the Drinfeld center$${\mathcal {Z}}({\mathcal {C}})$$of$${\mathcal {C}}$$adapted for$${\mathcal {M}}$$; we refer to this category as thereflective center$${\mathcal {E}}_{\mathcal {C}}({\mathcal {M}})$$of$${\mathcal {M}}$$. Just like$${\mathcal {Z}}({\mathcal {C}})$$is a canonical braided monoidal category attached to$${\mathcal {C}}$$, we show that$${\mathcal {E}}_{\mathcal {C}}({\mathcal {M}})$$is a canonical braided module category attached to$${\mathcal {M}}$$; its properties are investigated in detail. Our second goal pertains to when$${\mathcal {C}}$$is the category of modules over a quasitriangular Hopf algebraH, and$${\mathcal {M}}$$is the category of modules over anH-comodule algebraA. We show that the reflective center$${\mathcal {E}}_{\mathcal {C}}({\mathcal {M}})$$here is equivalent to a category of modules over an explicit algebra, denoted by$$R_H(A)$$, which we call thereflective algebraofA. This result is akin to$${\mathcal {Z}}({\mathcal {C}})$$being represented by the Drinfeld double$${\operatorname {Drin}}(H)$$ofH. We also study the properties of reflective algebras. Our third set of results is also in the Hopf setting above. We show that reflective algebras are quasitriangularH-comodule algebras, and we examine their corresponding quantumK-matrices; this yields solutions to the qRE. We also establish that the reflective algebra$$R_H(\mathbb {k})$$is an initial object in the category of quasitriangularH-comodule algebras, where$$\mathbb {k}$$is the ground field. The case whenHis the Drinfeld double of a finite group is illustrated.
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Braiding groups of automorphisms and almost-automorphisms of trees
Abstract We introduce “braided” versions of self-similar groups and Röver–Nekrashevych groups, and study their finiteness properties. This generalizes work of Aroca and Cumplido, and the first author and Wu, who considered the case when the self-similar groups are what we call “self-identical.” In particular, we use a braided version of the Grigorchuk group to construct a new group called the “braided Röver group,” which we prove is of type$$\operatorname {\mathrm {F}}_\infty $$. Our techniques involve using so-calledd-ary cloning systems to construct the groups, and analyzing certain complexes of embedded disks in a surface to understand their finiteness properties.
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- Award ID(s):
- 2343739
- PAR ID:
- 10627556
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Canadian Journal of Mathematics
- Volume:
- 76
- Issue:
- 2
- ISSN:
- 0008-414X
- Page Range / eLocation ID:
- 555 to 593
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.4153/S0008414X23000159
