quality:corruption, linguistic:voice_change
Commutativity as vector space operation

The tensor product of two vector spaces V {\displaystyle V} and W {\displaystyle W} is commutative in the sense that there is a canonical isomorphism:

that maps v ⊗ w {\displaystyle v\otimes w} to w ⊗ v {\displaystyle w\otimes v}.

On the other hand, even when V = W {\displaystyle V=W}, the tensor product of vectors is not commutative; that is v ⊗ w ≠ w ⊗ v {\displaystyle v\otimes w\neq w\otimes v}, in general.

The map x ⊗ y ↦ y ⊗ x {\displaystyle x\otimes y\mapsto y\otimes x} from V ⊗ V {\displaystyle V\otimes V} to itself induces a linear automorphism that is called a braiding map. More generally and as usual (see tensor algebra), let V ⊗ n {\displaystyle V^{\otimes n}} denote the tensor product of n copies of the vector space V. For every permutation s of the first n positive integers, the map:

induces a linear automorphism of V ⊗ n → V ⊗ n {\displaystyle V^{\otimes n}\to V^{\otimes n}}, which is called a braiding map.