dorsal/arxiv
View SchemaSquared Hopf algebras and reconstruction theorems
| Authors | Volodymyr V. Lyubashenko |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9605035 |
| URL | https://arxiv.org/abs/q-alg/9605035 |
| Journal | Proc. Workshop ``Quantum Groups and Quantum Spaces'' (Warszawa), Banach Center Publ., no. 40, Inst. Math. Polish Acad. Sci. (1997) 111--137 |
Abstract
Given an abelian k-linear rigid monoidal category V, where k is a perfect field, we define squared coalgebras as objects of cocompleted V tensor V (Deligne's tensor product of categories) equipped with the appropriate notion of comultiplication. Based on this, (squared) bialgebras and Hopf algebras are defined without use of braiding. If V is the category of k-vector spaces, squared (co)algebras coincide with conventional ones. If V is braided, a braided Hopf algebra can be obtained from a squared one. Reconstruction theorems give equivalence of squared co- (bi-, Hopf) algebras in V and corresponding fibre functors to V (which is not the case with other definitions). Finally, squared quasitriangular Hopf coalgebra is a solution to the problem of defining quantum groups in braided categories.
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"abstract": "Given an abelian k-linear rigid monoidal category V, where k is a perfect\nfield, we define squared coalgebras as objects of cocompleted V tensor V\n(Deligne\u0027s tensor product of categories) equipped with the appropriate notion\nof comultiplication. Based on this, (squared) bialgebras and Hopf algebras are\ndefined without use of braiding.\n If V is the category of k-vector spaces, squared (co)algebras coincide with\nconventional ones. If V is braided, a braided Hopf algebra can be obtained from\na squared one.\n Reconstruction theorems give equivalence of squared co- (bi-, Hopf) algebras\nin V and corresponding fibre functors to V (which is not the case with other\ndefinitions). Finally, squared quasitriangular Hopf coalgebra is a solution to\nthe problem of defining quantum groups in braided categories.",
"arxiv_id": "q-alg/9605035",
"authors": [
"Volodymyr V. Lyubashenko"
],
"categories": [
"q-alg",
"math.QA"
],
"journal_ref": "Proc. Workshop ``Quantum Groups and Quantum Spaces\u0027\u0027 (Warszawa),\n Banach Center Publ., no. 40, Inst. Math. Polish Acad. Sci. (1997) 111--137",
"title": "Squared Hopf algebras and reconstruction theorems",
"url": "https://arxiv.org/abs/q-alg/9605035"
},
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