Let $H$ be a Hopf algebra and $V$ a Yetter-Drinfeld module over $H$. Then there is a braiding $\Psi: V \otimes V \to V \otimes V$ given by $\Psi(x \otimes y) = x_{(-1)}.y \otimes x_{(0)}$, where $x_{(-1)} \otimes x_{(0)} = \delta(x)$. Let $T(V)$ be the tensor algebra of $V$.

Consider the multiplication on $T(V) \otimes T(V)$ defined as follows. For $a,b,c,d \in T(V)$, $(a \otimes b)(c \otimes d) = a \Psi(b \otimes c) d$.

The algebra $T(V)$ has a braided comultiplication $\Delta: T(V) \to T(V) \otimes T(V)$ such that $\Delta$ is an algebra map and $\Delta(v) = 1 \otimes v + v \otimes 1$. For example, for $x, y \in T(V)$, we have \begin{align} \Delta(xy) = 1 \otimes xy + xy \otimes 1 + x \otimes y + x_{(-1)}.y \otimes x_{(0)}. \end{align}

A primitive element $x$ in $T(V)$ is called primitive if $\Delta(x) = 1 \otimes x + x \otimes 1$. What are all primitive elements in $T(V)$?

Thank you very much.

Edit: the braiding $\Psi$ should be on $T(V)$: $\Psi: T(V) \otimes T(V) \to T(V) \otimes T(V)$.