(Vector space : Affine space) :: (Group : ???) & Principal Bundles
mark...@yahoo.com
As per the subject header, this question will be resolved here.
An obvious motivation for considering this issue is the awkwardness of
the formalism for principal bundles, where it is required that a base
space be fibred with a group, yet the identity of the group be treated
equivalently with the other elements. This is a clear indication that
an inappropriate formalism has been used for the theory, whereas a more
appropriate formalism would make use of the "affine" equivalent of a
group.
The characterization is simple. An "affine" group is a space G with a
ternary operation a/b.c satisfying the properties
(1) a/a.b = b
(2) a/b.b = a
(3) a/b.(c/d.e) = (a/b.c)/d.e
It is abelian if a/b.c = c/b.a.
This is to be understood as the group operation a/b.c = a b^{-1} c.
The corresponding definition of a homomorphism is then simply a map f:
G -> G' such that f(a/b.c) = f(a)/f(b).f(c).
For each element E of G, one may localize G to a group G_E with the
operations
ab = a/E.b
a^{-1} = E/a.E,
and readily prove that this is a group. The group is abelian iff the
"affine" group is abelian.
The induced map f: G_E -> G'_{f(E)} is then a gorup homomorphism if and
only if f is an "affine" group homomorphism.
An intrinsic characterization may be arrived at by considering the
relation
(a, b/c.d) = (c/b.a, d)
over G x G. Defining equvialence classes by [(a,b)] = a\b, one arrives
at an algebra Group(G) subject to the identity a\(b/c.d) = (c/b.a)\d.
This provides a suitable definition for the product (a\b)(c\d). With
respect to this product, one may prove that a\a is the identity, for
all a, and that the inverse of a\b is b\a.
The isomorphism of Group(G) and G_E is established with the maps a\b
|-> a/b.E and inverse map g |-> E\g. This gh = g/E.h -> E\(g/E.h) =
(E\g)(E\h), g^{-1} = E/g.E -> E\(E/g.E) = (E\E)(g\E) = (g\E) =
(E\g)^{-1}, and E -> E\E.
For a principal bundle P over a base space M with gauge group g =
Group(G), an alternate definition is then to simply define the space as
the cartesian product P = M x G, with the following items
(1) projection pi(m,a) = m
(2) group action (m,a) b\c = (m,a/b.c)
(3) "inner quotient" (m,a)\(m,b) = a\b.
The group action is proven to satisfy the properties pe = p; p(gh) =
(pg)h; the inner quotient, the properties p(p\q) = q and p\(pg) = g,
for elements p of P, q of {p}xG, g, h of Group(G).
To arrive at a principal bundle, one needs to impose a differential
structure, where it is already assumed that G and M are manifolds, with
the operation (a,b,c) |-> a/b.c smooth. (One can then call G a "Lie
affine group").
A "gauge" is a partial map S: M -> G with domain dom(S) open in M. Two
gauges S, S' are said to be compatible if S\S' is differentiable over
dom(S) intersect dom(S'). A principal bundle is then alternatively
defined as the cartesian product of a manifold M and Lie affine group G
which is covered by a family of mutually compatible gauges.
Maarten Bergvelt
In article <e4hnbc$fi$1...@dizzy.math.ohio-state.edu>, mark...@yahoo.com wrote:
>
> As per the subject header, this question will be resolved here.
If "this question" is the fill-in-the-blanks exercise, I think you are
looking for the word torsor, see for instance
http://math.ucr.edu/home/baez/torsors.html
--
Maarten Bergvelt