Reference request: compatibility conditions of four versions of Yetter-Drinfeld modules

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.

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.