Braided vector space

In mathematics, a braided vectorspace ${\displaystyle \;V}$ is a vector space together with an additional structure map ${\displaystyle \tau \;}$ symbolizing interchanging of two vector tensor copies:

such that the Yang–Baxter equation is fulfilled. Hence drawing tensor diagrams with ${\displaystyle \tau \;}$ an overcrossing the corresponding composed morphism is unchanged when a Reidemeister move is applied to the tensor diagram and thus they present a representation of the braid group.

As first example, every vector space is braided via the trivial braiding (simply flipping). A superspace has a braiding with negative sign in braiding two odd vectors. More generally, a diagonal braiding means that for a ${\displaystyle \;V}$-base ${\displaystyle x_{i}\;}$ we have

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