**Matti Pitkanen** (*matpitka@pcu.helsinki.fi*)

*Mon, 5 Apr 1999 08:08:31 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 171] Re: [time 165] Prime numbers in pregeometry"**Previous message:**Matti Pitkanen: "[time 169] Re: [time 160] Re: [time 157] tangent-cotangent; spaces that is!"**In reply to:**Stephen P. King: "[time 160] Re: [time 157] tangent-cotangent; spaces that is!"**Next in thread:**Ben Goertzel: "[time 159] Re: [time 149] Re: [time 108] Re: [time 81] Entropy, wholeness, dialogue, algebras"

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

*> Matti,
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*>
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*> Let me add: if we have tangent and cotangent spaces, do we also have
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*> algebras and "coalgebras" that are the relations among the points in the
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*> spaces? Can we have a 'hyperalgebra' of relations that maps algebras to
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*> coalgebras, and hyperhyperalgebras mapping between hyperalgebras? If so,
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*> we are speaking to what Bohm was trying to formalize with Implicate
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*> Orders and Baez with his N-Categories! Also this is the key idea used by
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*> Wegner in his study of interactive computing!
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*> 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,
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*>
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*> Stephen
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*>
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*> "Stephen P. King" wrote:
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*> >
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*> > Matti,
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*> >
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*> > Is there a cotangent space here? What relations would exist between the
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*> > tangent and cotangent spaces?
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*> >
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*> > I am not familiar with the meaning, e.g. I think visually, of what your
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*> > reply meams. :(
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*> >
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*> > Stephen
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*> >
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*> > Matti Pitkanen wrote:
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*> > >
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*> > > On Sun, 4 Apr 1999, Stephen P. King wrote:
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*> > >
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*> > > > Matti,
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*> > > >
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*> > > > Matti Pitkanen wrote:
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*> > > > >
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*> > > > snip
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*> > > >
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*> > > > > There might be something deep in induction of imbedding space
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*> > > > > tangent space octonion structure to spacetime surface [octonion units
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*> > > > > are projected to spacetime and their products which contain also
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*> > > > > part normal to surface are projected to spacetime surface so that one
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*> > > > > obtains tangent space projection C alpha beta gamma of structure constant
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*> > > > > tensor Cklm defined by IkIl = Ckl^mIm ]. But I do not know any idea about
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*> > > > > what deep consequences this might have. Quaternions appear
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*> > > > > in the construction of exact solutions of YM action (instantons): could
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*> > > > > octonions appear in the construction of the absolute minima of Kahler
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*> > > > > action if this construction is possible at all (just a free
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*> > > > > association(;-)?
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*> > > >
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*> > > > Is there a cotangent space here? What relations would exist between the
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*> > > > tangent and cotangent spaces?
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*> > >
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*> > > I_k can be regarded as 1-forms and since metric tensor
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*> > > is present one can map I_k to vector fields I^k by index raising.
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*> > >
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*> > > I_k is obtained from 'free' octonionic units I_A satisfying standard
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*> > > octonionic multiplication table by contracting with octobein e^A_k
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*> > >
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*> > > I_k= e^A_k I_A and this induces structure constant tensor
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*> > >
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*> > > Ckl^m= e^A_ke^B_ke^Cm C_ABC
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*> > >
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*> > > Metric is clearly essentially involved and one moves freely between forms
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*> > > and vectors.
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*> > >
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*> > > MP
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*>
*

**Next message:**Matti Pitkanen: "[time 171] Re: [time 165] Prime numbers in pregeometry"**Previous message:**Matti Pitkanen: "[time 169] Re: [time 160] Re: [time 157] tangent-cotangent; spaces that is!"**In reply to:**Stephen P. King: "[time 160] Re: [time 157] tangent-cotangent; spaces that is!"**Next in thread:**Ben Goertzel: "[time 159] Re: [time 149] Re: [time 108] Re: [time 81] Entropy, wholeness, dialogue, algebras"

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