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 ${V}^{*}={\mathrm{Hom}}_{k}\left(V,k\right)$ be the dual vector space and $\left({V}^{*}{\right)}^{\otimes m}$ be some tensor of ${V}^{*}$. Then one may define $\left(l,s\right)$-contraction

$\left({V}^{*}{\right)}^{\otimes m}\otimes {V}^{\otimes n}\to \left({V}^{*}{\right)}^{\otimes \left(m-1\right)}\otimes {V}^{\otimes \left(n-1\right)}$(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 $\left({V}^{*}{\right)}^{\otimes r}$ and $s$-th tensor factor of ${V}^{\otimes n}$. In fact as a map written, one can contract also elements of $\left({V}^{*}{\right)}^{\otimes m}\otimes {V}^{\otimes n}$ 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 $T\in \left({V}^{*}{\right)}^{\otimes r}$ is given in the dual basis by components ${T}_{{j}_{1},\dots ,{j}_{r}}$, then the components of the contraction will be

${\mathrm{contr}}_{l,s}\left(T,S{\right)}_{{j}_{1},\dots ,{j}_{l-1},{j}_{l+1},\dots ,{j}_{m}}^{{i}_{1},\dots ,{i}_{s-1},{i}_{s+1},\dots ,{i}_{n}}=\sum _{u=1}^{r}{T}_{{j}_{1},\dots ,{j}_{l-1},u,{j}_{l+1},\dots ,{j}_{m}}{S}^{{i}_{1},\dots ,{i}_{s-1},u,{i}_{s+1},\dots ,{i}_{n}}$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:

${\mathrm{contr}}_{l,s}\left(A{\right)}_{{j}_{1},\dots ,{j}_{l-1},{j}_{l+1},\dots ,{j}_{m}}^{{i}_{1},\dots ,{i}_{s-1},{i}_{s+1},\dots ,{i}_{n}}\right):=\sum _{u=1}^{r}{A}_{{j}_{1},\dots ,{j}_{l-1},u,{j}_{l+1},\dots ,{j}_{m}}^{{i}_{1},\dots ,{i}_{s-1},u,{i}_{s+1},\dots ,{i}_{n}}$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 $\left(1,1\right)$-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 $\omega \in \Lambda {V}^{*}$:

$\left(X,\omega \right)↦{\iota }_{X}\left(\omega \right)$(X,\omega)\mapsto \iota_X(\omega)

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

References

Revised on March 24, 2012 16:30:48 by Mike Shulman (71.136.234.110)