What is a Hopf copairing?

Question

I am trying to understand what Hopf copairing means.

From the definition of a Hopf pairing we know that it is a bilinear form of two Hopf algebras $H$ and $K$, $(-,-): H \times K \rightarrow \mathbb{K}$, such that for any $h,g \in H$ and $x,y \in K$, we have:

(1) $(h,xy)=\sum(h_{1},x)(h_{2},y)$,

(2) $(hg,x)=\sum(h,x_{1})(g,x_{2})$,

(3) $(1,x)=\epsilon(x)$, $(h,1)=\epsilon(h)$,

(4) $(h, Sx)=(Sh,x)$.

How can we define a copairing of Hopf algebras and what does it mean?

Thank you in advance! :)

Answer 1

I'm assuming that you are using Sweedler notations, and I will denote $(A \otimes B)^{12}=A \otimes B \otimes 1$, similarly for $(A \otimes B)^{13}$ and $(A \otimes B)^{23}$. (There is an ambiguity for 1, it should be seen as the unit of H or K according to the context.)

I have found this definition for a Hopf Algebra Copairing:

A map $ \mathbb{K} \to H \otimes K $ such that $1 \mapsto R= \sum R^1 \otimes R^2$ is a Hopf Copairing if :

1. $\sum \Delta(R^1) \otimes R^2 = R^{13} R^{23}$
2. $\sum R^1 \otimes \Delta(R^2) = R^{13} R^{12} $
3. $\sum \epsilon(R^1) R^2 =1_K$
4. $ \sum R^1 \epsilon(R^2) =1_A$

Is it relevant to you?

As for the meaning, I do not know how to interpret it, but I can give you some examples.

If $K=H$, and $R$ is an R-matrix of H, then it is also a copairing.
If $ K=H^{\star}$, and H finite dimensional, $\sum e_i \otimes e_i^{\star}$ is a copairing, where $e_i$ is a basis of H and $e_i^{\star}$ its dual basis.

I guess that if you have a pairing on H, K, it should induce a copairing on $H^{\star}, K^{\star}$. And conversely.