Threefold Way
The discussion at PF has taken a more octonion bent with the arrival of G. Dixon on the scene. Thanks also to Tony Smith for another interesting link.
Aside: All essential software installations complete. Cheap internet connection procured. Am now web surfing and working on applications at home.
Aside: All essential software installations complete. Cheap internet connection procured. Am now web surfing and working on applications at home.
posted by Kea | 6:20 PM
7 Comments:
Indeed it has. I especially like the mention of (OxO)P^2, a space that remains fairly mysterious partly because calculations are quite brutal over the octooctonions. Of course, we aren't in the stone age, so this can be remedied by writing an octooctonions class in some high-level programming language and having the ol' CPU carry out the grunt work.
Ultimately, (OxO)P^2 is an extension of OP^2, with points that are generalized projectors. I don't know how to define these yet, but taking the hint from the octonionic case, such matrices may be solutions to a generalized characteristic equation.
kneemo mentions (OxO)P2
so
I would like to recommend a very nice book by Boris Rosenfeld for whom (OxO)P2 is named as "Rosenfeld's Elliptic Projective Plane):
Geometry of Lie Groups (Kluwer 1997).
He has a web page at
http://www.math.psu.edu/katok_s/BR/init.html
I don't know how active he is on the web, but any insights he might have would probably be invaluable.
Tony Smith
Thank you both for the comments. I'm not sure what form of 'generalisation' you have in mind here, kneemo, but I'm guessing that categorical matrices might help.
Thanks for the link Tony. I just finished reading Rosenfeld's "Geometric Interpretations of Some Jordan Algebras", in which E6, E7 and E8 arise as groups of isometries of Hermitian elliptic planes whose 2-bichains are constructed from 4x4 Jordan matrices over H, CxH and HxH.
Now I see your motivation for using J(4,H) to elucidate F-theory.
Tony, I notice that at PF you mentioned the quaternion/real dichotomy coming from Clifford algebra for +++- and ---+ signatures. Naturally, this reminds me of Mulase et al's T duality in ribbon graph matrix models, which interchanges H and R via a simple transformation.
Mulase and Waldron in their 2002 paper "DUALITY OF ORTHOGONAL AND SYMPLECTIC MATRIX
INTEGRALS AND QUATERNIONIC FEYNMAN GRAPHS"
look at "... Graphical expansions of gauge theories ... recogniz[ing] that SO(2N)-gauge theory and Sp(N)-gauge theory are identical in their graphical expansions, except that the parameter N in the SO(2N)-theory has to be replaced with −N ..."
and
"... develop a graphical expansion technique for an N×N self-adjoint quaternionic matrix integral, and directly verify its duality with a real symmetric matrix integral of size 2N ...".
Such dualities can be seen from many points of view, which is a reason that I think that the underlying structures are really fundamental in physics as well as math.
When I look at the Mulase-Waldron type duality involving SO(2N) and Sp(N),
I am reminded of a duality between
the SO(2N+1) Lie algebra denoted by B_N
and
the Sp(N) Lie algebra denoted by C_N
that can be seen in terms of their root vectors, which are the structures taken as fundamental in Garrett Lisi's E8 physics model.
Here are some quotes from N. Bourbaki "Groupes et algebres de Lie" Chapitres 4, 5, et 6 (with slightly modified notation):
"... Planche II
Systeme de Type B_N (N at least 2)
... Racines [root vectors]: +/- e_i (i from 1 to N),
+/- e_i +/- e_j (i less than j and i,j from 1 to N)
... W(R) [the Weyl group] est produit semi-direct du [symmetric] groupe S_N ... et du groupe (Z/2Z)^N ...
Son ordere est 2^N . N!
... Matrice de Cartan (NxN)
...
Planche III
Systeme de Type C_N (N at least 2)
... Racines [root vectors]: +/- 2 e_i (i from 1 to N),
+/- e_i +/- e_j (i less than j and i,j from 1 to N)
... W(R) [the Weyl group] est produit semi-direct du [symmetric] groupe S_N ... et du groupe (Z/2Z)^N ...
Son ordere est 2^N . N!
... Matrice de Cartan (NxN) ...".
As you can see, the B_N and C_N Lie algebras have the same number of root vectors and they differ only in terms of the lengths (short = 1 or long = 2) of those of the form +/- e_i
B_N = SO(2N+1) has N short root vectors +/- e_i
while
C_N = Sp(N) has N long roots +/- 2 e_i
so
there is a duality by mapping the B_N short root vectors to the C_N long root vectors.
Note that since B_N and C_N have the same Weyl group, you cannot use the Weyl group to distinguish between them, so in some sense the root vectors seem to be more useful physically than the Weyl group.
My question is,
since SO(2N+1) contains SO(2N)
whether
the Mulase-Waldron-type duality between SO(2N) and Sp(N)
is entirely equivalent to
the root vector duality between SO(2N+1) and Sp(N)
?
Tony Smith
Hi Tony. Since I like to play with the ribbons and root lattices directly, it would be very nice to find their T duality purely in terms of lattice vectors, and a bonus to find a further link between Garrett's construction and string theory.
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