# [time 170] Re: [time 161] Re: [time 160] Re: [time 157] tangent-cotangent; spaces and algebras that is!

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Mon, 5 Apr 1999 08:08:31 +0300 (EET DST)

On Sun, 4 Apr 1999, Stephen P. King wrote:

> Matti,
>
> 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!
> http://www.cs.brown.edu/~pw/papers/math1.ps

Unfortunately, I do not have precise idea about what co-algebra means:
something like this I have met when trying to understand quantum groups
(with very little success!). So I must ask what co-algebra means.

The elements of octonionic algebra correspond to vector fields or
1-forms: X= X_kI^k =X^kI_k (sum understood), where X_k and X^k are
components of form/vector field. XY is the octonionic product of two
vector fields. As a matter fact, one could speak of local octonion
algebra. It is local number field (analogous to local gauge group). Local
number fields *almost* form an infinite dimensional number field: the
problems are caused by the zeros of X where 1/X explodes.

MP

> Onward,
>
> Stephen
>
> "Stephen P. King" wrote:
> >
> > Matti,
> >
> > 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
> >
> > Stephen
> >
> > 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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