Braided vector space
In mathematics, a braided vectorspace is a vector space together with an additional structure map symbolizing interchanging of two vector tensor copies:
such that the Yang–Baxter equation is fulfilled. Hence drawing tensor diagrams with 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. A superspace has a braiding with negative sign in braiding two odd vectors. More generally, a diagonal braiding means that for a -base we have
A good source for braided vector spaces entire braided monoidal categories with braidings between any objects, most importantly the modules over quasitriangular Hopf algebras and Yetter–Drinfeld modules over finite groups
If additionally possesses an algebra structure inside the braided category one has a braided commutator :
Examples of such braided algebras are the Nichols algebras, that are by definition generated by a given braided vectorspace. They appear as quantum Borel part of quantum groups and often possess an arithmetic root system, multiple Dynkin diagrams and a PBW-basis made up of braided commutators just like the ones in semisimple Lie algebras.
