Integrals and finite dimensionality in braided Hopf algebras

Question

Let $H$ be a Hopf algebra with invertible antipode. Let $A$ be a braided Hopf algebra in the Yetter-Drinfeld category ${}_H^H\mathcal{YD}$ over $H$.

A nonzero left integral in $A$ is a nonzero element $x\in A$ such that $yx=\epsilon(y)x$ for all $y\in A$. A nonzero right integral in $A$ is a nonzero element $x\in A$ such that $xy=\epsilon(y)x$ for all $y\in A$.

If $A$ is a classical Hopf algebra with trivial braiding, it is known that the existence of a nonzero left integral or nonzero right integral in $A$ implies finite dimensionality of $A$. This is a result due to Sweedler, Integrals for Hopf algebras, Annals of Mathematics, 1969.

My question concerns the braided case: Is the analogous result true for every braided Hopf algebra $A$? Does the existence of a nonzero left integral or nonzero right integral in $A$ imply finite dimensionality?

A reference for this general result would be welcome. If it helps, you could assume additionally that $A$ is a $\mathbb{Z}_{\geq 0}$-graded braided Hopf algebra, and if it helps further, you could assume additionally that $A$ is a connected $\mathbb{Z}_{\geq 0}$-graded braided Hopf algebra (i.e. $A^0=\mathbf{k}1$ where $\mathbf{k}$ is the ground field).

Remark. The proof of Sweedler for classical Hopf algebras does not directly carry over to the braided case.

Answer 1

I think the answer to your question is affirmative.

Consider the Radford biproduct $A\rtimes H$ defined in [1], which is an ordinary Hopf algebra over $\Bbbk$ defined on the vector space $A\otimes_{\Bbbk} H$. This construction does not require $A$ to be finite-dimensional over $\Bbbk$. The following was shown in [2, Section 4.6]: If $x$ is a non-zero left integral for $A$ and $\Lambda$ is a non-zero left integral for $H$, then $$\Lambda_{(1)}\cdot x\otimes \Lambda_{(2)}\in A\rtimes H$$ is a non-zero left integral of $A\rtimes H$. Hence, $A\rtimes H$ is finite-dimensional by the classical Larson--Sweedler result. In particular, $A$ is finite-dimensional over $\Bbbk$.

[1] Radford, David E., The structure of Hopf algebras with a projection, J. Algebra 92, 322-347 (1985). ZBL0549.16003.

[2] Burciu, Sebastian, A class of Drinfeld doubles that are ribbon algebras., J. Algebra 320, No. 5, 2053-2078 (2008). ZBL1163.16025.