1
$\begingroup$

There are four versions of compatibility conditions of Yetter-Drinfeld modules (left-left, left-right, right-left, right-right) in the article. Are there some references which derive these compatibility conditions? Thank you very much.

$\endgroup$
4
$\begingroup$

All of these conditions can be derived from Majid's construction of the dual of a monoidal category $\mathcal{C}$ [Maj1, Def. 3.2, 3.3] (in the reference, we want to consider the category denoted by $(\mathcal{C},F)^\circ$; see especially Expl. 3.4 which gives the monoidal center $\mathcal{Z}(\mathcal{C})$ as a special case of this more general construction; historical note: this construction was also known to Drinfeld before the publication of [Maj1] and used to construct the quantum double, but not published then; Drinfeld also observed that $\mathcal{Z}(\mathcal{C})$ is braided). What needs to be varied are two things:

  • The input category $\mathcal{C}$ with a monoidal functor $F\colon \mathcal{C} \to \mathcal{V}$ either to be

    (a) left or

    (b) right modules over a Hopf algebra $H$

either way, $F$ is the identity functor on $\mathcal{C}$.

  • In the definition of representations in [Maj1] (which are the objects of the category $(\mathcal{C},F)^\circ$ which gives the center) there are two possibilities:

    (i) One can consider a pair $V,\lambda$ with a natural isomorphism $\lambda_{V,X} \colon V\otimes F(K) \to F(K)\otimes V$, or, as it is invertible

    (ii) its inverse $\lambda_{V,X}^{-1}\colon F(K)\otimes V \to \colon V\otimes F(K)$. (Note: invertibility is automatic in the case where $H$ is a Hopf algebra with invertible antipode, i.e. the categories $(\mathcal{C},F)^\circ$ and $(\mathcal{C},F)^*$ defined in [Maj1] are the same)

The resulting category $(\mathcal{C},F)^\circ$ can be identified with Yetter--Drinfeld modules. The recipe is always as follows: Use $\lambda_{V,H}$ (respectively $\lambda_{V,H}^{-1}$ in (ii)), where $H$ has the regular action, and precompose with $\text{id}_V\otimes 1$ (respectively $1\otimes \text{id}_V$ in the case (ii)) to obtain a coaction. The requirent assument on $\lambda$ to be compatible with the tensor product (see Definition 3.2 in loc.cit.) translates to the thus obtained morphism to be a coaction. The requirement that $\lambda_{V,B}$ is a morphism of $H$-modules gives the Yetter--Drinfeld condition. To explicitly check this is a great exercise in Sweedler's notation calculus (or graphical calculus if one prefers).

One gets the following

  • YD-modules left action, left coaction: (a) + (i)
  • YD-modules left action, right coaction: (a) + (ii)
  • YD-modules right action, left coaction: (b) + (i)
  • YD-modules right action, right coaction: (b) + (ii)

I would be slightly hard pressed to find a place where this is proved in detail. I believe that with practice in typical calculations with Sweedler's notation, one can show this as an execise. If we think about quasi-Hopf algebras, a fairly explicit proof (however looking a comodules as input category $\mathcal{C}$) is given in [Maj2, Lemma 2.1]. Note that Hopf algebra are in particular quasi-Hopf algebras with trivial 3-cycle $\phi=1\otimes 1\otimes 1$, so the proof in [Maj2] only simplifies then.

References:

[Maj1] Majid, S.: Representations, duals and quantum doubles of monoidal categories. Proceedings of the Winter School on Geometry and Physics (Srní, 1990). Rend. Circ. Mat. Palermo (2) Suppl. No. 26 (1991), 197--206.

[Maj2] Majid, S. Quantum double for quasi-Hopf algebras. Lett. Math. Phys. 45 (1998), no. 1, 1--9.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.