nForum - Search Results Feed (Tag: tensors) 2021-12-03T05:45:36-05:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher tensor hierarchy https://nforum.ncatlab.org/discussion/10934/ 2020-02-12T10:05:17-05:00 2020-02-12T10:05:17-05:00 jim_stasheff https://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. ]]>
Contraction as (co)derivation https://nforum.ncatlab.org/discussion/10299/ 2019-09-06T10:14:58-04:00 2019-09-06T10:14:58-04:00 jim_stasheff https://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 $\omega\in \Lambda V^*$:&lt;latex&gt;(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? ]]>