Abstract index notation (also referred to as slot-naming index notation) is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeholders, not related to any basis and, in particular, are non-numerical. Thus it should not be confused with the Ricci calculus. The notation was introduced by Roger Penrose as a way to use the formal aspects of the Einstein summation convention to compensate for the difficulty in describing contractionss and covariant differentiation in modern abstract tensor notation, while preserving the explicit covariance of the expressions involved.

be a vector space its dual space. Consider, for example, an order-2 covariant can be identified with a bilinear form which can be represented as a pair of slots:

Abstract index notation is merely a labelling of the slots with Latin letters, which have no significance apart from their designation as labels of the slots (i.e., they are non-numerical):

A tensor contraction (or trace) between two tensors is represented by the repetition of an index label, where one label is contravariant (an upper index) and one label is covariant (a lower index). Thus, for instance,

over its last two slots. This manner of representing tensor contractions by repeated indices is formally similar to the Einstein summation convention. However, as the indices are non-numerical, it does not imply summation: rather it corresponds to the abstract basis-independent trace operation (or natural pairing.

Abstract indices and tensor spaces

A general homogeneous tensor is an element of a tensor product, such as

position. In this way, write the product as

or, simply

The last two expressions denote the same object as the first. Tensors of this type are denoted using similar notation, for example:

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,

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

and the second by

Braiding

To any tensor product on a single vector space, there are associated braiding maps. For example, the braiding map

). In general, the braiding maps are in one-to-one correspondence with elements of the symmetric group (represented as a product of disjoint cyclic permutations).

Braiding maps are important in differential geometry, for instance, in order to express the Bianchi identity denote the Riemann tensor. 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

Antisymmetrization and symmetrization

A general tensor may be antisymmetrized or symmetrized, and there is according notation.

is the symmetric group on three elements.

Similarly, we may symmetrize:

See also

• Penrose graphical notation

• Einstein notation

• Index notation

• Tensor

• Antisymmetric tensor

• Raising and lowering indices

• Covariance and contravariance of vectors

References