Abstract index notation

A general homogeneous tensor is an element of a tensor product of copies of V and V^*, such as :V \otimes V^* \otimes V^* \otimes V \otimes V^*. Label each factor in this tensor product with a Latin letter in a raised position for each contravariant V factor, and in a lowered position for each covariant V^* position. In this way, write the product as :V^a V_b V_c V^d V_e or, simply :V^a{}_{bc}{}^d{}_e. The last two expressions denote the same object as the first. Tensors of this type are denoted using similar notation, for example: :h^a{}_{bc}{}^d{}_e \in V^a{}_{bc}{}^d{}_e = V \otimes V^* \otimes V^* \otimes V \otimes V^*. == Contraction ==

In general, whenever one contravariant and one covariant factor occur in a tensor product of spaces, there is an associated contraction (or trace) map. For instance, : \mathrm{Tr}_{12} : V \otimes V^* \otimes V^* \otimes V \otimes V^* \to V^* \otimes V \otimes V^* is the trace on the first two spaces of the tensor product.\mathrm{Tr}_{15} : V \otimes V^* \otimes V^* \otimes V \otimes V^* \to V^* \otimes V^* \otimes V is the trace on the first and last space. These trace operations are signified on tensors by the repetition of an index. Thus the first trace map is given by : \mathrm{Tr}_{12} : h{}^a{}_b{}_c{}^d{}_e \mapsto h{}^a{}_a{}_c{}^d{}_e and the second by : \mathrm{Tr}_{15} : h{}^a{}_b{}_c{}^d{}_e \mapsto h{}^a{}_b{}_c{}^d{}_a. == Braiding ==

To any tensor product on a single vector space, there are associated braiding maps. For example, the braiding map : \tau_{(12)} : V \otimes V \rightarrow V \otimes V interchanges the two tensor factors (so that its action on simple tensors is given by \tau_{(12)} (v \otimes w) = w \otimes v). In general, the braiding maps are in one-to-one correspondence with elements of the symmetric group, acting by permuting the tensor factors. Here, \tau_\sigma denotes the braiding map associated to the permutation \sigma (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 R denote the Riemann tensor, regarded as a tensor in V^* \otimes V^* \otimes V^* \otimes V. The first Bianchi identity then asserts that :R + \tau_{(123)}R + \tau_{(132)}R = 0. 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 :R = R_{abc}{}^d \in V_{abc}{}^d = V^* \otimes V^* \otimes V^* \otimes V, the Bianchi identity becomes :R_{abc}{}^d + R_{cab}{}^d + R_{bca}{}^d = 0. == Antisymmetrization and symmetrization ==

A general tensor may be antisymmetrized or symmetrized, and there is according notation. We demonstrate the notation by example. Let's antisymmetrize the type-(0,3) tensor \omega_{abc}, where \mathrm{S}_3 is the symmetric group on three elements. : \omega_{[abc]} := \frac{1}{3!} \sum_{\sigma \in \mathrm{S}_3} {\text{sgn}(\sigma)} \omega_{\sigma(a)\sigma(b)\sigma(c)} Similarly, we may symmetrize: : \omega_{(abc)} := \frac{1}{3!} \sum_{\sigma \in \mathrm{S}_3} \omega_{\sigma(a)\sigma(b)\sigma(c)} == See also ==