I've been trying to find some cross(?) product in 5-space. I found something... I haven't worked out an example yet,... it might be laborious.
In Cl<sub>5,0</sub> of R<sup>5</sup> (It is proper to say the first space here is 'of' the second?), we have:
1 5 10 10 5 1
for the dimension (in one of the other senses of dimension...) of the various multivector spaces... scalar, vector, bivector,... etc.
I'm trying to find a cross(?) product of some nature between the bivectors and trivectors. I think these two spaces may be isomorphic, and if so a Hodge dual should be able to be set up...? (I think...)
One reason I'm not really sure if the two spaces are isomorphic,... and also perhaps share a 1-to-1 correspondence between elements is because there are non-simple bivectors and trivectors in 5-space. I'm not sure if it's safe to say that there are only going to be <sup>10</sup> cardinality of elements in each of these spaces. (I think this probably means a direct product 10 's. I'm still not sure about this cardinality stuff... See first reply to original post here) I'm concerned that even if we can expect this cardinality of elements (whatever precisely it might mean, or tell us...) that the non-simple elements might throw this cardinality out of whack... I'm just not sure...?
Anyway... so far this is what I'm come up with. It isn't really a relation between the bivectors or trivectors of 5-space per se, but rather between 3 1-vectors and bivectors... although one question I have is whether this is essentially still between bivectors or trivectors since wedging 3 1-vectors produces a trivector...? However, even if that's essentially accurate, I think it's always going to be a simple trivector, so we'll not get all 5-space trivectors this way.
I think I also found a simpler formula for resolving permutations like these which so often come up in various product calculations... See last reply by me in post Minimum transpositions of permutations...?
I'll summarize my questions here:
- Are the spaces of bivectors and trivectors in 5-space isomorphic?
- Do these two spaces have the same cardinality of elements? The non-simple elements don't alter the cardinality? What cardinality do the simple elements, non-simple, elements comprise?
- Can a Hodge duality be set up between these two spaces?
- Does what I've derived here qualify as a 'cross' product?
- If yes to the last question, will the resulting bivector here be perpendicular to each of the 3 1-vector operands? Can these 3 1-vector operands also be seen as a trivector?