3 ms·
> There are no interesting formulas in a tensor algebra, because the whole idea of it being “most general” (every other algebra is a quotient) is that it has no
by pjbk 6y ago
> There are no interesting formulas in a tensor algebra, because the whole idea of it being “most general” (every other algebra is a quotient) is that it has no relations. It is only when passing to more specific algebras (symmetric, exterior, GA, ...) that the subject really becomes interesting.
And yet you can still go a level further, with objects like those in Moon & Spencer's holor theory that include generalizations for transformation rules beyond covariant or contravariant, generalized connections similar --for example-- to universal Christoffel symbols, and effectively creating rules for tensors, pseudotensors and stranger array-like objects.
- JadeNB 6y ago> And yet you can still go a level further, with objects like those in Moon & Spencer's holor theory that include generalizations for transformation rules beyond covariant or contravariant, generalized connections similar --for example-- to universal Christoffel symbols, and effectively creating rules for tensors, pseudotensors and stranger array-like objects. One can always go farther, but I'm not sure that this is doing that. I looked up the Wiki page https://en.wikipedia.org/wiki/Parry_Moon#Holors https://en.wikipedia.org/wiki/Parry_Moon#Holors and found: > But this is incorrect, in general, for the definition of a tensor includes a specific dependence on coordinate transformation. This is very much a physicist's definition of a tensor; a mathematician's definition includes no such thing (although one may prove results about co-ordinate transformations as a theorem). For example, there is no such thing, mathematically, as an intrinsically covariant or contravariant tensor, although, having chosen a fixed vector space $V$, one may certainly put a bi-grading on $T(V \otimes V^*)$ that makes that sort of distinction.