8
$\begingroup$

It is known that the Drinfel'd double $D(A)$ of a Hopf algebra $A$ is characterized by the following two properties:

  1. The category of left $D(A)$-modules $_{D(A)}\mathcal{M}$ is equivalent to the category of Yetter-Drinfeld modules $_A\mathcal{YD}^A$.
  2. The category of left $D(A)$-modules $_{D(A)}\mathcal{M}$ is equivalent to the center $\mathcal{Z}(_A\mathcal{M})$ of $_A\mathcal{M}$.

In general, for a (possibly degenerate) skew-Hopf pairing $\eta:A\otimes B \to k$ one can define a quantum double $D(A,B,\eta)$, s.t. the Drinfel'd double is the special case $B=A^*$ (see [KS], 8.2.1). Is there a categorical interpretation of $D(A,B,\eta)$ in analogy to $D(A)$?

[KS] Klymik, Schmüdgen - Hopf algebras and their representations

EDIT: My first guess is that $_{D(A,B,\eta)}\mathcal{M}$ is equivalent to the category of both left $A$-modules and right $B$-modules $M$, satisfying $$\eta(a_{(1)},b_{(1)})(a_{(2)}.m).b_{(2)}=\eta(a_{(2)},b_{(2)})a_{(1)}.(m.b_{(1)})$$

Improve this question
$\endgroup$

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.