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I am interested in knowing if the category $^H_H\mathcal{YD}$ of left-left Yetter-Drinfeld modules over an infinite dimensional Hopf algebra $H$ is an abelian category or not.

The answer is affirmative if $H$ is finite-dimensional, because in such a case $^H_H\mathcal{YD}$ would be equivalent to the category of modules over the Drinfeld double of $H$.

Surprisingly, I didn't found an answer googling the question for the infinite-dimensional case, whence I considered asking it here.

Question 1: Is the category $^H_H\mathcal{YD}$ of Yetter-Drinfeld modules over an infinite dimensional Hopf algebra $H$ is an abelian category?

Question 2: (which extends Question 1, in fact) Is the Drinfeld center of an abelian monoidal category still abelian, in general?

My feeling is that both answers should be positive and that they should be well-known. If anybody is aware of it, would he be so kindly to add also a reference to the literature, please?

Many thanks.

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question 1: yes. Kernel and cokernel of a morphism is the same as kernel and cokernel of it as linear map, becaause a Kernel of a module map is a submodule and a kernel of a comodule map is subcomodule, idem cokernel. Also a module and comodule map that is an isomorphism as linear map, is an isomorphism as module and comodule map.

question 2: Yes, the center of a category is formed by pairs $(X, \Phi)$ where $X$ is an object and $\Phi$ a natural transformation between $X\otimes (-) $ and $(-)\otimes X$. For a map between pairs $(X,\Phi)\to (Y,\Psi)$, take the object kernel of the map $X\to Y$, and check that the natural isomorphism between $X\otimes (-) $ and $(-)\otimes X$ induce one between $Ker(X\to Y)\otimes (-) $ and $(-)\otimes Ker(X\to Y)$.

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  • $\begingroup$ Caveat: you need additional assumption to have that the category of comodules admits kernels. For example, it is true if the coalgebra is flat over the base commutative ring $\Bbbk$, but in general it is not (see e.g. "Corings and Comodules" by Brzezinski and Wisbauer). Nevertheless, I found a reference which says exactly that under the flatness assumption everything goes smoothly: Caenepeel, Militaru, Zhu, "Frobenius and Separable Functors for Generalized Module Categories and Nonlinear Functions", Corollary 30. Concerning the Drinfeld center, I will check if there are or not flatness issues $\endgroup$ – Ender Wiggins May 25 at 16:58
  • $\begingroup$ BTW, do you have any reference for your claim? $\endgroup$ – Ender Wiggins May 25 at 16:59

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