nLab tensor contraction

Contraction of tensors

By tensor we mean a vector in some tensor power $V^{\otimes n}$ of a vector $k$-space $V$ (or a projective $k$-module if $k$ is only a commutative ring). Let $V^* = Hom_k(V,k)$ be the dual vector space and $(V^*)^{\otimes m}$ be some tensor of $V^*$. Then one may define $(l,s)$-contraction

$$(V^*)^{\otimes m}\otimes V^{\otimes n}\to (V^*)^{\otimes (m-1)}\otimes V^{\otimes (n-1)}$$

by pairing by the evaluation map the $l$-th tensor factor of $(V^*)^{\otimes r}$ and $s$-th tensor factor of $V^{\otimes n}$. In fact as a map written, one can contract also elements of $(V^*)^{\otimes m}\otimes V^{\otimes n}$ which did not come from a product of a pair of element (i.e. which are not decomposable tensors).

Let the rank $r$ of $V$ be finite. If $S\in V^{\otimes n}$ is given in some basis by components $S^{i_1,\ldots, i_n}$ and $T\in (V^*)^{\otimes r}$ is given in the dual basis by components $T_{j_1,\ldots,j_r}$, then the components of the contraction will be

$$contr_{l,s}(T,S)^{i_1,\ldots, i_{s-1},i_{s+1},\ldots,i_n}_{j_1,\ldots, j_{l-1},j_{l+1},\ldots,j_m} = \sum_{u = 1}^r T_{j_1,\ldots, j_{l-1},u,j_{l+1},\ldots,j_m} S^{i_1,\ldots, i_{s-1},u,i_{s+1},\ldots,i_n}$$

More generally one can contract mixed tensors and do several contractions simultaneously. In fact it is better to think of a contraction as a tensor multiplication of two tensors and then doing the genuine contraction of one upper and one lower index of the same tensor:

$$contr_{l,s}(A)^{i_1,\ldots, i_{s-1},i_{s+1},\ldots,i_n}_{j_1,\ldots,j_{l-1},j_{l+1},\ldots, j_m}) \;\coloneqq\; \sum_{u = 1}^r A^{i_1,\ldots, i_{s-1},u,i_{s+1},\ldots,i_n}_{j_1,\ldots,j_{l-1},u,j_{l+1},\ldots, j_m}$$

The simplest case is the trace of a $(1,1)$-tensor: $tr A = \sum_{i=1}^r A^i_i$.

These operations can be symmetrized or antisymmetrised appropriately to make sense on symmetric or antisymmetric powers.

For example, there is a contraction of a vector $X\in V$ and a $n$-form $\omega\in \Lambda V^*$:

$$(X,\omega)\mapsto \iota_X(\omega)$$

and $\iota_X: \omega\mapsto \iota_X(\omega)$ is a graded derivation of the exterior algebra of degree $-1$. This is also done for the tangent bundle which is a $C^\infty(M)$-module $V = T M$, then one gets the contraction of vector fields and differential forms. It can also be done in vector spaces, fibrewise.

Related concepts

• tensor algebra

• symmetric algebra

• exterior algebra

• divided power algebra

• multilinear algebra

References

• Wikipedia: Tensor contraction

• Shlomo Sternberg: Introduction to differential geometry