# Squared Hopf algebras and reconstruction theorems

Banach Center Publications (1997)

- Volume: 40, Issue: 1, page 111-137
- ISSN: 0137-6934

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topLyubashenko, Volodymyr. "Squared Hopf algebras and reconstruction theorems." Banach Center Publications 40.1 (1997): 111-137. <http://eudml.org/doc/252188>.

@article{Lyubashenko1997,

abstract = {Given an abelian 𝑉-linear rigid monoidal category 𝑉, where 𝑉 is a perfect field, we define squared coalgebras as objects of cocompleted 𝑉 ⨂ 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 𝑉-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 the usual definitions). Finally, squared quasitriangular Hopf coalgebra is a solution to the problem of defining quantum groups in braided categories.},

author = {Lyubashenko, Volodymyr},

journal = {Banach Center Publications},

keywords = {rigid monoidal categories; tensor functors; tensor products; tensor squares; squared coalgebras; bicoalgebras; reconstruction theories; quantum groups; braided categories; squared quasitriangular Hopf coalgebras},

language = {eng},

number = {1},

pages = {111-137},

title = {Squared Hopf algebras and reconstruction theorems},

url = {http://eudml.org/doc/252188},

volume = {40},

year = {1997},

}

TY - JOUR

AU - Lyubashenko, Volodymyr

TI - Squared Hopf algebras and reconstruction theorems

JO - Banach Center Publications

PY - 1997

VL - 40

IS - 1

SP - 111

EP - 137

AB - Given an abelian 𝑉-linear rigid monoidal category 𝑉, where 𝑉 is a perfect field, we define squared coalgebras as objects of cocompleted 𝑉 ⨂ 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 𝑉-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 the usual definitions). Finally, squared quasitriangular Hopf coalgebra is a solution to the problem of defining quantum groups in braided categories.

LA - eng

KW - rigid monoidal categories; tensor functors; tensor products; tensor squares; squared coalgebras; bicoalgebras; reconstruction theories; quantum groups; braided categories; squared quasitriangular Hopf coalgebras

UR - http://eudml.org/doc/252188

ER -

## References

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- [2] V. G. Drinfeld, phQuantum groups, Proceedings of the ICM, AMS, Providence, R.I. 1 (1987), 798-820.
- [3] A. Grothendieck and J. L. Verdier, phPréfaisceuax, in: Théorie des topos et cohomologie étale des schémas (SGA 4), Lect. Notes Math. 269, Berlin, Heidelberg, New York: Springer-Verlag, 1972, 1-217.
- [4] L. Hlavaty, phQuantized braided groups, J. Math. Phys. 35 (1994), no. 5, 2560-2569. Zbl0810.17007
- [5] S. MacLane, phCategories for the Working Mathematician, Springer-Verlag, 1971.
- [6] B. Pareigis, phReconstruction of hidden symmetries, preprint 1994.
- [7] N. Saavedra Rivano, phCatégories Tannakiennes, Lect. Notes Math. 265, Berlin, Heidelberg, New York: Springer-Verlag, 1972.
- [8] P. Schauenburg, phTannaka duality for Arbitrary Hopf Algebras, Algebra-Berichte 66, München: R. Fisher, 1992.

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