nForum - Search Results Feed (Tag: tensors)2021-12-03T05:45:36-05:00https://nforum.ncatlab.org/
Lussumo Vanilla & Feed Publisher
tensor hierarchyhttps://nforum.ncatlab.org/discussion/10934/2020-02-12T10:05:17-05:002020-02-12T10:05:17-05:00jim_stasheffhttps://nforum.ncatlab.org/account/12/
The n-lab has no entry for `tensor hierarchy'. The terminology has not quite stabilized. I see it referred to mostly along with the embedding tensor.It would be good to agree on a definition.
The n-lab has no entry for `tensor hierarchy'. The terminology has not quite stabilized. I see it referred to mostly along with the embedding tensor. It would be good to agree on a definition.
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Contraction as (co)derivationhttps://nforum.ncatlab.org/discussion/10299/2019-09-06T10:14:58-04:002019-09-06T10:14:58-04:00jim_stasheffhttps://nforum.ncatlab.org/account/12/
The n-lab is very clear on contraction as a derivationFor example, there is a contraction of a vector $X\in V$ and a $n$-form $\omega\in \Lambda V^*$:<latex>(X,\omega)\mapsto ...
The n-lab is very clear on contraction as a derivation For example, there is a contraction of a vector $X\in V$ and a $n$-form $\omega\in \Lambda V^*$:

<latex>(X,\omega)\mapsto \iota_X(\omega)</latex>

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.

Is it written somewhere about contraction of a $1$-form $\omega$ with an$n-vector $X\in \in \Lambda V$ as a coderivation?
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