**Abstract index notation** is a mathematical notation for

tensorTensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

s and

spinorIn mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...

s that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeholders, not related to any fixed basis and, in particular, are non-numerical. The notation was introduced by

Roger PenroseSir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...

as a way to use the formal aspects of the Einstein summation convention in order to compensate for the difficulty in describing

contractionIn multilinear algebra, a tensor contraction is an operation on one or more tensors that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor caused by applying the summation...

s and

covariant differentiationIn mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...

in modern abstract tensor notation, while preserving the explicit covariance of the expressions involved.
Let

*V* be a vector space, and

*V*^{*} its dual. Consider, for example, a rank-2

covariantIn multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis from one coordinate system to another. When one coordinate system is just a rotation of the other, this...

tensor

$\backslash scriptstyle\; h\backslash in\; V^*\backslash otimes\; V^*$. Then

*h* can be identified with a

bilinear form on

*V*. In other words, it is a function of two arguments in

*V* which can be represented as a pair of

*slots*:

$h\; =\; h(-,-).\backslash ,$
Abstract index notation is merely a

*labelling* of the slots by Latin letters, which have no significance apart from their designation as labels of the slots (i.e., they are non-numerical):
NEWLINE

NEWLINE- $h\; =\; h\_\{ab\}.\backslash ,$

NEWLINE
A contraction between two tensors is represented by the repetition of an index label, where one label is contravariant (an

*upper index* corresponding to a tensor in

*V*) and one label is covariant (a

*lower index* corresponding to a tensor in

*V*^{*}). Thus, for instance,
NEWLINE

NEWLINE- $\{t\_\{ab\}\}^b$

NEWLINE
is the trace of a tensor

*t* =

*t*_{ab}^{c} 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 duality pairing) between tensor factors of type

*V* and those of type

*V*^{*}.

## Abstract indices and tensor spaces

A general homogeneous tensor is an element of a

tensor productIn mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...

of copies of

*V* and

*V*^{*}, such as

$V\backslash otimes\; V^*\backslash otimes\; V^*\; \backslash otimes\; V\backslash otimes\; V^*.$
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

$V^a\; V\_b\; V\_c\; V^d\; V\_e\backslash ,$
or, simply

$$**Abstract index notation** is a mathematical notation for

tensorTensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

s and

spinorIn mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...

s that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeholders, not related to any fixed basis and, in particular, are non-numerical. The notation was introduced by

Roger PenroseSir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...

as a way to use the formal aspects of the Einstein summation convention in order to compensate for the difficulty in describing

contractionIn multilinear algebra, a tensor contraction is an operation on one or more tensors that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor caused by applying the summation...

s and

covariant differentiationIn mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...

in modern abstract tensor notation, while preserving the explicit covariance of the expressions involved.
Let

*V* be a vector space, and

*V*^{*} its dual. Consider, for example, a rank-2

covariantIn multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis from one coordinate system to another. When one coordinate system is just a rotation of the other, this...

tensor

$\backslash scriptstyle\; h\backslash in\; V^*\backslash otimes\; V^*$. Then

*h* can be identified with a

bilinear form on

*V*. In other words, it is a function of two arguments in

*V* which can be represented as a pair of

*slots*:

$h\; =\; h(-,-).\backslash ,$
Abstract index notation is merely a

*labelling* of the slots by Latin letters, which have no significance apart from their designation as labels of the slots (i.e., they are non-numerical):
NEWLINE

NEWLINE- $h\; =\; h\_\{ab\}.\backslash ,$

NEWLINE
A contraction between two tensors is represented by the repetition of an index label, where one label is contravariant (an

*upper index* corresponding to a tensor in

*V*) and one label is covariant (a

*lower index* corresponding to a tensor in

*V*^{*}). Thus, for instance,
NEWLINE

NEWLINE- $\{t\_\{ab\}\}^b$

NEWLINE
is the trace of a tensor

*t* =

*t*_{ab}^{c} 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 duality pairing) between tensor factors of type

*V* and those of type

*V*^{*}.

## Abstract indices and tensor spaces

A general homogeneous tensor is an element of a

tensor productIn mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...

of copies of

*V* and

*V*^{*}, such as

$V\backslash otimes\; V^*\backslash otimes\; V^*\; \backslash otimes\; V\backslash otimes\; V^*.$
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

$V^a\; V\_b\; V\_c\; V^d\; V\_e\backslash ,$
or, simply

$$**Abstract index notation** is a mathematical notation for

tensorTensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

s and

spinorIn mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...

s that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeholders, not related to any fixed basis and, in particular, are non-numerical. The notation was introduced by

Roger PenroseSir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...

as a way to use the formal aspects of the Einstein summation convention in order to compensate for the difficulty in describing

contractionIn multilinear algebra, a tensor contraction is an operation on one or more tensors that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor caused by applying the summation...

s and

covariant differentiationIn mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...

in modern abstract tensor notation, while preserving the explicit covariance of the expressions involved.
Let

*V* be a vector space, and

*V*^{*} its dual. Consider, for example, a rank-2

covariantIn multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis from one coordinate system to another. When one coordinate system is just a rotation of the other, this...

tensor

$\backslash scriptstyle\; h\backslash in\; V^*\backslash otimes\; V^*$. Then

*h* can be identified with a

bilinear form on

*V*. In other words, it is a function of two arguments in

*V* which can be represented as a pair of

*slots*:

$h\; =\; h(-,-).\backslash ,$
Abstract index notation is merely a

*labelling* of the slots by Latin letters, which have no significance apart from their designation as labels of the slots (i.e., they are non-numerical):
NEWLINE

NEWLINE- $h\; =\; h\_\{ab\}.\backslash ,$

NEWLINE
A contraction between two tensors is represented by the repetition of an index label, where one label is contravariant (an

*upper index* corresponding to a tensor in

*V*) and one label is covariant (a

*lower index* corresponding to a tensor in

*V*^{*}). Thus, for instance,
NEWLINE

NEWLINE- $\{t\_\{ab\}\}^b$

NEWLINE
is the trace of a tensor

*t* =

*t*_{ab}^{c} 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 duality pairing) between tensor factors of type

*V* and those of type

*V*^{*}.

## Abstract indices and tensor spaces

A general homogeneous tensor is an element of a

tensor productIn mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...

of copies of

*V* and

*V*^{*}, such as

$V\backslash otimes\; V^*\backslash otimes\; V^*\; \backslash otimes\; V\backslash otimes\; V^*.$
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

$V^a\; V\_b\; V\_c\; V^d\; V\_e\backslash ,$
or, simply

$\{\{\{V^a\}\_\{bc\}\}^d\}\_e.$
It is important to remember that these last two expressions signify precisely the same object as the first. We shall denote tensors of this type by the same sort of notation, for instance

$\{\{\{h^a\}\_\{bc\}\}^d\}\_e\; \backslash in\; \{\{\{V^a\}\_\{bc\}\}^d\}\_e\; =\; V\backslash otimes\; V^*\backslash otimes\; V^*\; \backslash otimes\; V\backslash otimes\; V^*.$
## 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,

$\backslash mathrm\{Tr\}\_\{12\}\; :\; V\backslash otimes\; V^*\backslash otimes\; V^*\; \backslash otimes\; V\backslash otimes\; V^*\; \backslash to\; V^*\; \backslash otimes\; V\backslash otimes\; V^*$
is the trace on the first two spaces of the tensor product.

$\backslash mathrm\{Tr\}\_\{15\}\; :\; V\backslash otimes\; V^*\backslash otimes\; V^*\; \backslash otimes\; V\backslash otimes\; V^*\; \backslash to\; V^*\; \backslash otimes\; V^*\backslash otimes\; V$
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

$\backslash mathrm\{Tr\}\_\{12\}\; :\; \{\{\{h^a\}\_\{bc\}\}^d\}\_e\; \backslash mapsto\; \{\{\{h^a\}\_\{ac\}\}^d\}\_e$
and the second by

$\backslash mathrm\{Tr\}\_\{15\}\; :\; \{\{\{h^a\}\_\{bc\}\}^d\}\_e\; \backslash mapsto\; \{\{\{h^a\}\_\{bc\}\}^d\}\_a$
## Braiding

To any tensor product, there are associated

braiding maps. For example, the braiding map

$\backslash tau\_\{(12)\}\; :\; V\backslash otimes\; V\; \backslash rightarrow\; V\backslash otimes\; V$
interchanges the two tensor factors (so that its action on simple tensors is given by

$\backslash tau\; (v\; \backslash otimes\; w)\; =\; w\; \backslash otimes\; v$). In general, the braiding maps are in one-to-one correspondence with elements of the

symmetric groupIn mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...

, acting by permuting the tensor factors. Here, we use

$\backslash tau\_\backslash sigma$ to denote the braiding map associated to the permutation

$\backslash sigma$ (represented as a product of disjoint

cyclic permutationA cyclic permutation or circular permutation is a permutation built from one or more sets of elements in cyclic order.The notion "cyclic permutation" is used in different, but related ways:- Definition 1 :right|mapping of permutation...

s).
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

$V^*\; \backslash otimes\; V^*\; \backslash otimes\; V^*\; \backslash otimes\; V$. The first Bianchi identity then asserts that

$R+\backslash tau\_\{(123)\}R+\backslash tau\_\{(132)\}R\; =\; 0.$
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

$R=\{R\_\{abc\}\}^d\backslash in\; \{V\_\{abc\}\}^d\; =\; V^*\backslash otimes\; V^*\backslash otimes\; V^*\backslash otimes\; V,$
the Bianchi identity becomes

$\{R\_\{abc\}\}^d+\{R\_\{cab\}\}^d+\{R\_\{bca\}\}^d\; =\; 0.$