Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sun, 4 Apr 1999 20:50:14 +0300 (EET DST)
On Sun, 4 Apr 1999, Stephen P. King wrote:
> Matti,
>
> Matti Pitkanen wrote:
> >
> snip
>
> > There might be something deep in induction of imbedding space
> > tangent space octonion structure to spacetime surface [octonion units
> > are projected to spacetime and their products which contain also
> > part normal to surface are projected to spacetime surface so that one
> > obtains tangent space projection C alpha beta gamma of structure constant
> > tensor Cklm defined by IkIl = Ckl^mIm ]. But I do not know any idea about
> > what deep consequences this might have. Quaternions appear
> > in the construction of exact solutions of YM action (instantons): could
> > octonions appear in the construction of the absolute minima of Kahler
> > action if this construction is possible at all (just a free
> > association(;-)?
>
> Is there a cotangent space here? What relations would exist between the
> tangent and cotangent spaces?
I_k can be regarded as 1-forms and since metric tensor
is present one can map I_k to vector fields I^k by index raising.
I_k is obtained from 'free' octonionic units I_A satisfying standard
octonionic multiplication table by contracting with octobein e^A_k
I_k= e^A_k I_A and this induces structure constant tensor
Ckl^m= e^A_ke^B_ke^Cm C_ABC
Metric is clearly essentially involved and one moves freely between forms
and vectors.
MP
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