nLab
contraction

Disambiguation

In physics, contraction is a dilation with coefficient $\lambda <1$. This notion is used in fixed point theory, theory of topological vector spaces etc. There is also a notion of contraction from metric space theory; see short map. Finally, the contraction rule is a structural rule in logic and type theory. This entry will be predominantly about another notion of a contraction.

Contraction of tensors

This entry will be predominantly about contraction of tensors, where 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 $V^{*}={\mathrm{Hom}}_k(V,k)$ be the dual vector space and $(V^{*}{)}^{\otimes m}$ be some tensor of $V^{*}$. Then one may define $(l,s)$-contraction

$$(V^{*}{)}^{\otimes m}\otimes V^{\otimes n}\to (V^{*}{)}^{\otimes (m-1)}\otimes V^{\otimes (n-1)}$$(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 $S^{i_1,\dots ,i_n}$ and $T\in (V^{*}{)}^{\otimes r}$ is given in the dual basis by components $T_{j_1,\dots ,j_r}$, then the components of the contraction will be

$${\mathrm{contr}}_{l,s}(T,S{)}_{j_1,\dots ,j_{l-1},j_{l+1},\dots ,j_m}^{i_1,\dots ,i_{s-1},i_{s+1},\dots ,i_n}=\sum _{u=1}^r{T}_{j_1,\dots ,j_{l-1},u,j_{l+1},\dots ,j_m}{S}^{i_1,\dots ,i_{s-1},u,i_{s+1},\dots ,i_n}$$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:

$${\mathrm{contr}}_{l,s}(A{)}_{j_1,\dots ,j_{l-1},j_{l+1},\dots ,j_m}^{i_1,\dots ,i_{s-1},i_{s+1},\dots ,i_n}):=\sum _{u=1}^r{A}_{j_1,\dots ,j_{l-1},u,j_{l+1},\dots ,j_m}^{i_1,\dots ,i_{s-1},u,i_{s+1},\dots ,i_n}$$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}) := \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: $\mathrm{tr}A={\sum }_{i=1}^r{A}_i^i$.

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 $\omega \in \Lambda V^{*}$:

$$(X,\omega )\mapsto {\iota }_X(\omega )$$(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 $C^{\mathrm{\infty }}(M)$-module $V=TM$, then one gets the contraction of vector fields and differential forms. It can also be done in vector spaces, fibrewise.

References

• wikipedia: tensor contraction

• Shlomo Sternberg, Introduction to differential geometry