In one of my courses we were given the definition of a braided vector space:
Let $k$ be a field. A braided vector space is a pair $(V,s)$ with
- $V$ a $k$-vector space
- $s:V \otimes V \rightarrow V \otimes V$ an invertible linear map which obeys the equation $(s \otimes id_V) \circ (id_V \otimes s) \circ (s \otimes id_V)= (id_V \otimes s) \circ (s \otimes id_V) \circ (id_V \otimes s)$.
- Why are these vector spaces any interesting? How should one think about them?
To me the definition comes out of the blue. My thoughts so far: The condition for $s$ appears to be something like the Reidemeister move type III. So, braided vector spaces seem like building blocks for linear representations of the braid group. Namely, given a braided vector space $(V,s)$ there is a group homomorphism $p_n: B_n \rightarrow Aut(V^{\otimes n}); \sigma _i \mapsto s_i$ for any positive integer $n$. Here, $\sigma_i$ denotes the i-th generator of the braid group, while $s_i$ refers to the automorphism on $V^{\otimes n}$ that „braids the i-th copy of the tensor product.“ (I could make it more precise, but then reading it would be tedious. If it is unclear, please ask).