2. Th e discrete Euler-Arnold equation (I)
(a) Algebras, groups and /^-matrices
Let G, g, 7r+, and 7r_ be as defined in Section 1. A convenient variable on E is given
by
* = (A-1)/( A + 1)
(2.1)
which takes
E - S ^ {|*| = 1}
{Re A 0 } -{\z\ 1}
{Re A 0 } - { | * | 1}
oo —• 1
d\/(l- A2) -+ -ctz/2z
and has inverse
A = (l + z ) / ( l - z ) .
Smooth loops on E are mapped to smooth loops on 5 1 ,
X(z)=
Yl
X^
(2.2)
.3)
with ||-X"j|| rapidly decreasing as \j\ —• oo. We will use both the A variable and the z
variable for the loops, depending on convenience. We will, however, consistently abuse
notation and denote X(X(z)) simply by X{z), etc.
Notatio n (2.4)
We use the standard notation §^{-)-~i t o denote the Cauchy principal value at z = 1,
=f A(z)^-=lim[ A(z)-^-,
J W 2 - l elO J\;-Ue 2 - 1
(2.5)
where the latter integral is taken counterclockwise.
12
Previous Page Next Page