Sunday, April 30, 2006

Cartan Form Exterior Super Algebra

Each element of the super-algebra is of the form

a(p+1) + db(p)

a is a p + 1 form

b is a p-form

d = Cartan exterior derivative that is nilpotent

d^2 = 0

Addition law

[a(p+1) + db(p)] + [c(p+1) + de(p)] = [a(p+1) + c(p+1)] + d[b(p) + e(p)]

Exterior multiplication law

[a(p+1) + db(p)]/\[c(p+1) + de(p)]

= a(p+1)/\c(p+1) + d[a(p+1)/\e(p) + b(p)/\c(p+1)]

In n-dim space the largest non-zero p-form is an n-form.

Is this well-known or did I just invent it?

Ref. p. 877 "The Road to Reality" by Roger Penrose

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