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.



Integrals and finite dimensionality in braided Hopf algebras
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**](http://dx.doi.org/10.1016/0021-8693(85)90124-3), J. Algebra 92, 322-347 (1985). [ZBL0549.16003](https://zbmath.org/?q=an:0549.16003).


[2] *Burciu, Sebastian*, [**A class of Drinfeld doubles that are ribbon algebras.**](http://dx.doi.org/10.1016/j.jalgebra.2008.05.021), J. Algebra 320, No. 5, 2053-2078 (2008). [ZBL1163.16025](https://zbmath.org/?q=an:1163.16025).


