The following example of a braided vector space is given in my lecture notes:
Let $k$ be a field, and $n>1$. Let $V$ be an $n$-dimensional vector space with ordered basis $(e_1, e_2,..., e_n)$. Let $q \in k$ be invertible. Define $s \in End(V \otimes V)$: $$s(e_i \otimes e_j) = \begin{cases} q \: e_i \otimes e_j & i=j \\ e_j \otimes e_i & i < j \\ e_j \otimes e_i + (q - q^{-1}) e_i \otimes e_j & j< i.\end{cases}$$ Then $(V, s)$ is a braided vector space.
My questions:
- How do you come up with the above example? Are there examples in a similar vein? In my lecture notes it says that this example is a one-parameter deformation of another example, namely $(V, t)$ with the twist map $t: e_i \otimes e_j \mapsto e_j \otimes e_i$. It seems, this exploits the canonical surjective group homomorphism $B_n \rightarrow S_n$. However, I am not exactly sure how it does so.
- I suppose you prove that $(V,s)$ is a braided vector space (i.e. checking Yang-Baxter, and taking determinant of the respective transformation matrix) via induction over $n$. Correct?