Advanced book on Mathematics Olympiad

2.4 Abstract Algebra 89

Becauseα(X)∗α(Y )=−α(Y )∗α(X), this implies thatα(X)∗α(Y )=0, and conse-
quentlyα(X+Y)=α(X)+α(Y ). In particular,αis additive on every one-dimensional
space, whenceα(rX)=rα(X), for every rational numberr. Butαis continuous, so
α(sX)=sα(X)for every real numbers. Applying this property we find that for any
X, Y∈R^3 ands∈R,

sα

(

X+Y+

1

2

sX∗Y

)

=α

(

sX+sY+

1

2

s^2 X∗Y

)

=α(sX)◦(sY ))

=α(sX)◦α(sY )=(sα(X))◦(sα(Y ))

=sα(X)+sα(Y )+

1

2

s^2 α(X)∗α(Y ).

Dividing both sides bys, we obtain

α

(

X+Y+

1

2

sX∗Y

)

=α(X)+α(Y )+

1

2

sX∗Y.

In this equality if we lets→0, we obtainα(X+Y)=α(X)+α(Y ). Also, if we let
s=1 and use the additivity we just proved, we obtainα(X∗Y)=α(X)∗α(Y ). The
problem is solved. 

Traditionally,X∗Y is denoted by[X, Y]andR^3 endowed with this operation is
called the Heisenberg Lie algebra. Also,R^3 endowed with◦is called the Heisenberg
group. And we just proved a famous theorem showing that a continuous automorphism
of the Heisenberg group is also an automorphism of the Heisenberg Lie algebra. The
Heisenberg group and algebra are fundamental concepts of quantum mechanics.

267.With the aid of a calculator that can add, subtract, and determine the inverse of
a nonzero number, find the product of two nonzero numbers using at most 20
operations.

268.Invent a binary operation from which+,−,×, and/can be derived.

269.A finite setSis endowed with an associative binary operation∗that satisfies(a∗
a)∗b=b∗(a∗a)=bfor alla, b∈S. Prove that the set of all elements of the
forma∗(b∗c)witha, b, cdistinct elements ofScoincides withS.

270.LetSbe the smallest set of rational functions containingf (x, y)=xandg(x, y)=
yand closed under subtraction and taking reciprocals. Show thatSdoes not contain
the nonzero constant functions.

271.Let∗and◦be two binary operations on the setM, with identity elementse, respec-
tively,e′, and with the property that for everyx, y, u, v∈M,
(x∗y)◦(u∗v)=(x◦u)∗(y◦v).
Prove that