nLab tensor contraction



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id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


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Contraction of tensors

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

(V *) mV n(V *) (m1)V (n1)(V^*)^{\otimes m}\otimes V^{\otimes n}\to (V^*)^{\otimes (m-1)}\otimes V^{\otimes (n-1)}

by pairing by the evaluation map the ll-th tensor factor of (V *) r(V^*)^{\otimes r} and ss-th tensor factor of V nV^{\otimes n}. In fact as a map written, one can contract also elements of (V *) mV n(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 rr of VV be finite. If SV nS\in V^{\otimes n} is given in some basis by components S i 1,,i nS^{i_1,\ldots, i_n} and T(V *) rT\in (V^*)^{\otimes r} is given in the dual basis by components T j 1,,j rT_{j_1,\ldots,j_r}, then the components of the contraction will be

contr l,s(T,S) j 1,,j l1,j l+1,,j m i 1,,i s1,i s+1,,i n= u=1 rT j 1,,j l1,u,j l+1,,j mS i 1,,i s1,u,i s+1,,i ncontr_{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) j 1,,j l1,j l+1,,j m i 1,,i s1,i s+1,,i n):= u=1 rA j 1,,j l1,u,j l+1,,j m i 1,,i s1,u,i s+1,,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)(1,1)-tensor: trA= i=1 rA i itr 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 XVX\in V and a nn-form ωΛV *\omega\in \Lambda V^*:

(X,ω)ι X(ω)(X,\omega)\mapsto \iota_X(\omega)

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


Last revised on April 27, 2023 at 15:12:15. See the history of this page for a list of all contributions to it.