Is $T(V)$ an almost cocommutative algebra?

Question

Let $T(V)$ be an algebra and $T(V)$ its tensor algebra. The algebra $T(V)$ is a Hopf algebra: https://en.wikipedia.org/wiki/Tensor_algebra. My question is: is $T(V)$ an almost cocommutative Hopf algebra? Here a Hopf algebra $A$ is almost cocommutative if its comultiplication and its opposite comultiplication differ by conjugation by a distinguished invertible element $R \in A \otimes A$: $$ \tau \Delta(a) = R\Delta(a)R^{-1}. $$

Answer 1

If we consider the tensor algebra $T(V)$ equipped with the coproduct defined by $$ \Delta(x)=1\otimes x+x\otimes 1 $$ for all elements $x\in T^1(V)=V$ and homomorphically extended to the whole of $T(V)$, then it is cocommutative and since cocommutative are a subset of almost cocommutative Hopf algebras, $T(V)$ is almost cocommutative -and even quasitriangular- in a trivial way through $R=1 \otimes 1$.

Answer 2

In your case, $R$ would be an invertible element of $T(V)\otimes T(V)$. Can you find all invertible elements in the algebra $T(V)\otimes T(V)$? There are not a lot of them...