# Integrals and finite dimensionality in braided Hopf algebras

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.

Consider the Radford biproduct $$A\rtimes H$$ defined in , 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 $$\Bbbk$$.