ON THE COMPOSITION FUNCTIONS OF
NILPOTENT LIE GROUPS
KUO-TSAI CHEN
This note contains a proof of the theorem:
If the composition
functions of a real or complex Lie group are
polynomials, then it is nilpotent.
The converse theorem is found among E. Cartan's works [2] and
can be also shown through the Baker-Hausdorff
formula in a rather
straightforward
fashion (see [l]).
Let ® be a real or complex Lie group with Lie algebra g. Choose for
g a base Xi, • ■ • , Xn, dual to which there is a base wi, • • • , «, of
the Maurer-Cartan
forms. Denote by x = (xi, • • • , x„) the corresponding canonical coordinate system and by/;(x, y), i=\,
■ • ■ , n,
the composition functions with respect to the coordinate system. If
ft(x, y), i=\,
■ ■ ■ , n, are polynomials,
we are going to prove that
® is nilpotent.
For XE$, define R(X) to be the linear transformation
(operating
on the right side) of the vector space of g such that, for F(Eg,
YR(X) = [Y, X] = YX - XY.
Allow the vector space of g to be normed
so that the norm | R(X) \
of R(X) is defined.
It is known that,
forms
about
the identity
e of ®, the Maurer-Cartan
wj(x, dx) = ^2 An(x)dxj,
are explicitly
given through
i = 1, • • • , «,
the formulas
X) Ajk(x)Xi = Xk-——-1
ft = 1, • • • , «,
where X —2^XiXi.
We assert that, about e, the left invariant infinitesimal
transformations Xi= ^Hij(x)d/dxj,
i= 1, • • ■ , n, can be explicitly given
through the formulas
ZHtj(x)Xi=Xi-
expR(X) — 1
•
Presented to the Society, November 27, 1953 under the title On a Cartan's theorem
and its converse; received by the editors January 26, 1957.
1158
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COMPOSITIONFUNCTIONSOF N1LPOTENT LIE GROUPS
In fact, if Xi —^Hij(x)d/dxj,
by the preceding
£
equation,
^(Xj)Xi
i = 1, • • • , n, where the Hy are defined
then we have about e,
= £
Aik(x)Hjk(x)Xi
Vff
Since
the
series
1159
MY exp*(AQ-l
= Z H^)X„
R{x)
- y._
expit(X)-l
^
' exp R(X) - 1
R(X)
represented
by R(X)/(exp
respectively
R(X) — l)
and (exp R(X) —l)/R(X)
converge both absolutely,
we obtain
^c»)i(Xj)Xi = Xj, i.e. o)i(Xj)=8ij. Therefore each Xi coincides with
X, about e.
On the other hand, we observe that each Ha(x) is equal to
(dfi(x, y)/dyj)y=0 which is a polynomial, say, of degree ^N. Let
R(X)/(exp R(X) -1) be expanded in the series ^=0
set BpXi[R(X)]"=
^Hijp(x)Xj.
polynomial of degree p, and
Bp [R(X) ]" and
Then each Hijp(x) is a homogeneous
Hij(x) = ha + Z HijP(x).
p=i
We conclude that BpXi[R(X)]p = 0, i = l, ■ ■ ■, n, and therefore
BpY[R(X)]v
with.6,^0,
theorem
= 0 for any
F£g
when p>N.
Since there
exists q>N
this yields Y[R(X) ]« = 0 for any X, F£g. According to a
due to Engel (see [3]), g is nilpotent.
Hence ® is nilpotent.
Bibliography
1. G. Birkhoff, Analytical groups, Trans. Amer. Math. Soc. vol. 43 (1938) pp. 61-
101.
2. E. Cartan,
Les reprisentations
linlaires
des groupes de Lie, J. Math.
Pures
Appl. vol. 17 (1938) pp. 1-12.
3. H. Zassenhaus,
Uber Liesche Ringe mit Primzahlcharakteristik,
Sem. Hanischen Univ. vol. 13 (1939) pp. 1-100.
University
of Hong Kong
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