# What are all primitive elements in a tensor algebra?

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 $T(V)$: $\Psi: T(V) \otimes T(V) \to T(V) \otimes T(V)$.

• You have defined $\Psi$ on $V \otimes V$, but then you write $\Psi\left(b \otimes c\right)$ where $b, c \in T(V)$. WHat do you mean? – darij grinberg Nov 24 '16 at 10:25
• @darij grinberg, thank you very much. I edited the post. – Jianrong Li Nov 24 '16 at 11:12

Nevertheless, if you have $(V,\mathfrak{c})$ a braided vector space over a field $\Bbbk$, then $T_{\Bbbk}(V)$ (performed in the category of $\Bbbk$-vector spaces) inherits the structure of a graded braided bialgebra and its primitives are nicely described in Ardizzoni, On the combinatorial rank of a graded braided bialgebra, Lemma 2.7.
As far as I know (see also here), to have that the category of Yetter-Drinfeld modules over a Hopf algebra is braided you need bijectivity of the antipode. If this is the case, the sentence between Definition 2.1 and Definition 2.2 in the aforementioned paper ensures that your example falls into the picture above, i.e., the underlying vector space of theYetter-Drinfeld module $V$ is a braided vector space and its tensor algebra $T_{\Bbbk}(V)$ is a graded braided bialgebra.