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Max Demirdilek
Max Demirdilek
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1. Context.
1.1. Classical Yetter-Drinfeld modules.
Let $H$ a bialgebra in a braided monoidal category $\mathcal{C}$. A left-right Yetter-Drinfeld module over $H$ is a triple $(V,\rho,\Delta)$ consisting of a left $H$-module $(V,\rho)$ and a right $H$-comodule $(V,\Delta)$ satisfying the Yetter-Drinfeld condition. Morphisms of left-right Yetter-Drinfeld modules are morphisms of left modules and right comodules. The category ${}_H{\mathcal{YD}^H}$ of left-right Yetter-Drinfeld modules over $H$ can be given the structure of a braided monoidal category such that the forgetful functor to $\mathcal{C}$ is a monoidal equivalence. If $\mathcal{C}$ is the category of $k$-vector spaces and $H$ is a finite-dimensional Hopf algebra, then there are braided monoidal equivalences $${}_H{\mathcal{YD}^H}\cong \mathcal{Z}(H\text{-mod})\cong D(H)\text{-mod},$$ where $\mathcal{Z}(H\text{-mod})$ denotes the Drinfeld center of the category of left $H$-modules and $D(H)$ is the Drinfeld double of $H$. In other words, the category of (left-right) Yetter-Drinfeld modules realizes the braided monoidal category $\mathcal{Z}(H\text{-mod})$ directly in terms of the Hopf algebra $H$.

1.2. Hopf monads.
Bruguières' and Virelizier's Hopf monads generalize Hopf algebras to the non-braided setting. Given a centralizable Hopf monad $T$ on a rigid monoidal category $\mathcal{C}$, one can introduce its double Hopf monad $D_T$ (see Section 6.2.) and then find a braided monoidal isomorphism $\mathcal{Z}(T\mathcal{-C})\cong D_T\mathcal{-C}$, where $\mathcal{Z}(T\mathcal{-C})$ denotes the Drinfeld center of the category of modules over the monad $T$ and $D_T\mathcal{-C}$ is the category of modules over the monad $T$$D_T$. Hopf modules also straight-forwardly generalize to the Hopf monad setting; see the 2006 paper by Bruguières and Virelizier.

2. Question.
Is there a analogous notion of Yetter-Drinfeld module in the setting of Hopf monads? Even less precisely: Can the braided monoidal category $\mathcal{Z}(T\mathcal{-C})$ be realized directly in terms of the Hopf monad $T$?

1. Context.
1.1. Classical Yetter-Drinfeld modules.
Let $H$ a bialgebra in a braided monoidal category $\mathcal{C}$. A left-right Yetter-Drinfeld module over $H$ is a triple $(V,\rho,\Delta)$ consisting of a left $H$-module $(V,\rho)$ and a right $H$-comodule $(V,\Delta)$ satisfying the Yetter-Drinfeld condition. Morphisms of left-right Yetter-Drinfeld modules are morphisms of left modules and right comodules. The category ${}_H{\mathcal{YD}^H}$ of left-right Yetter-Drinfeld modules over $H$ can be given the structure of a braided monoidal category such that the forgetful functor to $\mathcal{C}$ is a monoidal equivalence. If $\mathcal{C}$ is the category of $k$-vector spaces and $H$ is a finite-dimensional Hopf algebra, then there are braided monoidal equivalences $${}_H{\mathcal{YD}^H}\cong \mathcal{Z}(H\text{-mod})\cong D(H)\text{-mod},$$ where $\mathcal{Z}(H\text{-mod})$ denotes the Drinfeld center of the category of left $H$-modules and $D(H)$ is the Drinfeld double of $H$. In other words, the category of (left-right) Yetter-Drinfeld modules realizes the braided monoidal category $\mathcal{Z}(H\text{-mod})$ directly in terms of the Hopf algebra $H$.

1.2. Hopf monads.
Bruguières' and Virelizier's Hopf monads generalize Hopf algebras to the non-braided setting. Given a centralizable Hopf monad $T$ on a rigid monoidal category $\mathcal{C}$, one can introduce its double Hopf monad $D_T$ (see Section 6.2.) and then find a braided monoidal isomorphism $\mathcal{Z}(T\mathcal{-C})\cong D_T\mathcal{-C}$, where $\mathcal{Z}(T\mathcal{-C})$ denotes the Drinfeld center of the category of modules over the monad $T$ and $D_T\mathcal{-C}$ is the category of modules over the monad $T$. Hopf modules also straight-forwardly generalize to the Hopf monad setting; see the 2006 paper by Bruguières and Virelizier.

2. Question.
Is there a analogous notion of Yetter-Drinfeld module in the setting of Hopf monads? Even less precisely: Can the braided monoidal category $\mathcal{Z}(T\mathcal{-C})$ be realized directly in terms of the Hopf monad $T$?

1. Context.
1.1. Classical Yetter-Drinfeld modules.
Let $H$ a bialgebra in a braided monoidal category $\mathcal{C}$. A left-right Yetter-Drinfeld module over $H$ is a triple $(V,\rho,\Delta)$ consisting of a left $H$-module $(V,\rho)$ and a right $H$-comodule $(V,\Delta)$ satisfying the Yetter-Drinfeld condition. Morphisms of left-right Yetter-Drinfeld modules are morphisms of left modules and right comodules. The category ${}_H{\mathcal{YD}^H}$ of left-right Yetter-Drinfeld modules over $H$ can be given the structure of a braided monoidal category such that the forgetful functor to $\mathcal{C}$ is a monoidal equivalence. If $\mathcal{C}$ is the category of $k$-vector spaces and $H$ is a finite-dimensional Hopf algebra, then there are braided monoidal equivalences $${}_H{\mathcal{YD}^H}\cong \mathcal{Z}(H\text{-mod})\cong D(H)\text{-mod},$$ where $\mathcal{Z}(H\text{-mod})$ denotes the Drinfeld center of the category of left $H$-modules and $D(H)$ is the Drinfeld double of $H$. In other words, the category of (left-right) Yetter-Drinfeld modules realizes the braided monoidal category $\mathcal{Z}(H\text{-mod})$ directly in terms of the Hopf algebra $H$.

1.2. Hopf monads.
Bruguières' and Virelizier's Hopf monads generalize Hopf algebras to the non-braided setting. Given a centralizable Hopf monad $T$ on a rigid monoidal category $\mathcal{C}$, one can introduce its double Hopf monad $D_T$ (see Section 6.2.) and then find a braided monoidal isomorphism $\mathcal{Z}(T\mathcal{-C})\cong D_T\mathcal{-C}$, where $\mathcal{Z}(T\mathcal{-C})$ denotes the Drinfeld center of the category of modules over the monad $T$ and $D_T\mathcal{-C}$ is the category of modules over the monad $D_T$. Hopf modules also straight-forwardly generalize to the Hopf monad setting; see the 2006 paper by Bruguières and Virelizier.

2. Question.
Is there a analogous notion of Yetter-Drinfeld module in the setting of Hopf monads? Even less precisely: Can the braided monoidal category $\mathcal{Z}(T\mathcal{-C})$ be realized directly in terms of the Hopf monad $T$?

Source Link
Max Demirdilek
Max Demirdilek
  • 764
  • 4
  • 13

Yetter-Drinfeld modules for Hopf monads

1. Context.
1.1. Classical Yetter-Drinfeld modules.
Let $H$ a bialgebra in a braided monoidal category $\mathcal{C}$. A left-right Yetter-Drinfeld module over $H$ is a triple $(V,\rho,\Delta)$ consisting of a left $H$-module $(V,\rho)$ and a right $H$-comodule $(V,\Delta)$ satisfying the Yetter-Drinfeld condition. Morphisms of left-right Yetter-Drinfeld modules are morphisms of left modules and right comodules. The category ${}_H{\mathcal{YD}^H}$ of left-right Yetter-Drinfeld modules over $H$ can be given the structure of a braided monoidal category such that the forgetful functor to $\mathcal{C}$ is a monoidal equivalence. If $\mathcal{C}$ is the category of $k$-vector spaces and $H$ is a finite-dimensional Hopf algebra, then there are braided monoidal equivalences $${}_H{\mathcal{YD}^H}\cong \mathcal{Z}(H\text{-mod})\cong D(H)\text{-mod},$$ where $\mathcal{Z}(H\text{-mod})$ denotes the Drinfeld center of the category of left $H$-modules and $D(H)$ is the Drinfeld double of $H$. In other words, the category of (left-right) Yetter-Drinfeld modules realizes the braided monoidal category $\mathcal{Z}(H\text{-mod})$ directly in terms of the Hopf algebra $H$.

1.2. Hopf monads.
Bruguières' and Virelizier's Hopf monads generalize Hopf algebras to the non-braided setting. Given a centralizable Hopf monad $T$ on a rigid monoidal category $\mathcal{C}$, one can introduce its double Hopf monad $D_T$ (see Section 6.2.) and then find a braided monoidal isomorphism $\mathcal{Z}(T\mathcal{-C})\cong D_T\mathcal{-C}$, where $\mathcal{Z}(T\mathcal{-C})$ denotes the Drinfeld center of the category of modules over the monad $T$ and $D_T\mathcal{-C}$ is the category of modules over the monad $T$. Hopf modules also straight-forwardly generalize to the Hopf monad setting; see the 2006 paper by Bruguières and Virelizier.

2. Question.
Is there a analogous notion of Yetter-Drinfeld module in the setting of Hopf monads? Even less precisely: Can the braided monoidal category $\mathcal{Z}(T\mathcal{-C})$ be realized directly in terms of the Hopf monad $T$?