Stephen P. King (firstname.lastname@example.org)
Sun, 04 Apr 1999 14:32:02 -0400
Let me add: if we have tangent and cotangent spaces, do we also have
algebras and "coalgebras" that are the relations among the points in the
spaces? Can we have a 'hyperalgebra' of relations that maps algebras to
coalgebras, and hyperhyperalgebras mapping between hyperalgebras? If so,
we are speaking to what Bohm was trying to formalize with Implicate
Orders and Baez with his N-Categories! Also this is the key idea used by
Wegner in his study of interactive computing!
"Stephen P. King" wrote:
> Is there a cotangent space here? What relations would exist between the
> tangent and cotangent spaces?
> I am not familiar with the meaning, e.g. I think visually, of what your
> reply meams. :(
> Matti Pitkanen wrote:
> > 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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