Please use this identifier to cite or link to this item: http://hdl.handle.net/1959.14/145764
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- Title
- Hopf monads on monoidal categories
- Related
- Advances in mathematics, Vol. 227, No. 2, (2011), p.745-800
- DOI
- 10.1016/j.aim.2011.02.008
- Publisher
- Elsevier
- Date
- 2011
- Author/Creator
- Bruguières, Alain
- Author/Creator
- Lack, Steve
- Author/Creator
- Virelizier, Alexis
- Description
- We define Hopf monads on an arbitrary monoidal category, extending the definition given in Bruguières and Virelizier (2007) [5] for monoidal categories with duals. A Hopf monad is a bimonad (or opmonoidal monad) whose fusion operators are invertible. This definition can be formulated in terms of Hopf adjunctions, which are comonoidal adjunctions with an invertibility condition. On a monoidal category with internal Homs, a Hopf monad is a bimonad admitting a left and a right antipode.Hopf monads generalize Hopf algebras to the non-braided setting. They also generalize Hopf algebroids (which are linear Hopf monads on a category of bimodules admitting a right adjoint). We show that any finite tensor category is the category of finite-dimensional modules over a Hopf algebroid.Any Hopf algebra in the center of a monoidal category C gives rise to a Hopf monad on C. The Hopf monads so obtained are exactly the augmented Hopf monads. More generally if a Hopf monad T is a retract of a Hopf monad P, then P is a cross product of T by a Hopf algebra of the center of the category of T-modules (generalizing the Radford-Majid bosonization of Hopf algebras).We show that the comonoidal comonad of a Hopf adjunction is canonically represented by a cocommutative central coalgebra. As a corollary, we obtain an extension of Sweedler's Hopf module decomposition theorem to Hopf monads (in fact to the weaker notion of pre-Hopf monad).
- Description
- 56 page(s)
- Subject Keyword
- Cross products
- Subject Keyword
- Hopf algebras
- Subject Keyword
- Hopf algebroids
- Subject Keyword
- Hopf monads
- Subject Keyword
- Monoidal categories
- Resource Type
- journal article
- Organisation
- Macquarie University. Dept. of Mathematics
- Identifier
- http://hdl.handle.net/1959.14/145764
- Identifier
- ISSN:0001-8708
- Identifier
- mq_res-ext-2-s2.0-79953709685
- Language
- eng
- Reviewed
