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.

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