dorsal/arxiv
View SchemaCrossed modules and quantum groups in braided categories
| Authors | Yu. N. Bespalov |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9510013 |
| URL | https://arxiv.org/abs/q-alg/9510013 |
Abstract
Let $A$ be a Hopf algebra in a braided category $\cal C$. Crossed modules over $A$ are introduced and studied as objects with both module and comodule structures satisfying a compatibility condition. The category $\DY{\cal C}^A_A$ of crossed modules is braided and is a concrete realization of a known general construction of a double or center of a monoidal category. For a quantum braided group $(A,\overline A,{\cal R})$ the corresponding braided category of modules ${\cal C}_{\cO{A,\overline A}}$ is identified with a full subcategory in $\DY{\cal C}_A^A$. The connection with cross products is discussed and a suitable cross product in the class of quantum braided groups is built. Majid--Radford theorem, which gives equivalent conditions for an ordinary Hopf algebra to be such a cross product, is generalized to the braided category. Majid's bosonization theorem is also generalized.
{
"annotation_id": "f02e39cd-29e5-43ef-a415-67076909d854",
"date_created": "2026-03-02T18:01:25.096000Z",
"date_modified": "2026-03-02T18:01:25.096000Z",
"file_hash": "c2b4110fbe74e8837a861140da9230a24031478594e1fce8cf07293173f2c4ce",
"private": false,
"record": {
"abstract": "Let $A$ be a Hopf algebra in a braided category $\\cal C$. Crossed modules\nover $A$ are introduced and studied as objects with both module and comodule\nstructures satisfying a compatibility condition. The category $\\DY{\\cal C}^A_A$\nof crossed modules is braided and is a concrete realization of a known general\nconstruction of a double or center of a monoidal category. For a quantum\nbraided group $(A,\\overline A,{\\cal R})$ the corresponding braided category of\nmodules ${\\cal C}_{\\cO{A,\\overline A}}$ is identified with a full subcategory\nin $\\DY{\\cal C}_A^A$. The connection with cross products is discussed and a\nsuitable cross product in the class of quantum braided groups is built.\nMajid--Radford theorem, which gives equivalent conditions for an ordinary Hopf\nalgebra to be such a cross product, is generalized to the braided category.\nMajid\u0027s bosonization theorem is also generalized.",
"arxiv_id": "q-alg/9510013",
"authors": [
"Yu. N. Bespalov"
],
"categories": [
"q-alg",
"math.QA"
],
"title": "Crossed modules and quantum groups in braided categories",
"url": "https://arxiv.org/abs/q-alg/9510013"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "90a01656-2f8c-402e-9a1c-2a556c06349a",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}