Braiding
To any tensor product, there are associated braiding maps. For example, the braiding map
interchanges the two tensor factors (so that its action on simple tensors is given by ). In general, the braiding maps are in one-to-one correspondence with elements of the symmetric group, acting by permuting the tensor factors. Here, we use to denote the braiding map associated to the permutation (represented as a product of disjoint cyclic permutations).
Braiding maps are important in differential geometry, for instance, in order to express the Bianchi identity. Here let denote the Riemann tensor, regarded as a tensor in . The first Bianchi identity then asserts that
Abstract index notation handles braiding as follows. On a particular tensor product, an ordering of the abstract indices is fixed (usually this is a lexicographic ordering). The braid is then represented in notation by permuting the labels of the indices. Thus, for instance, with the Riemann tensor
the Bianchi identity becomes
Read more about this topic: Abstract Index Notation