36 2. Basic notions of representation theory

Moreover, it is easy to see that we have decompositions

SV =

n≥0

SnV,

∧V =

n≥0

∧nV.

2.13. Hilbert’s third problem

Problem 2.13.1. It is known that if A and B are two polygons of the

same area, then A can be cut by finitely many straight cuts into pieces

from which one can make B (check it — it is fun!). David Hilbert

asked in 1900 whether it is true for polyhedra in three dimensions. In

particular, is it true for a cube and a regular tetrahedron of the same

volume?

The answer is “no”, as was found by Dehn in 1901. The proof is

very beautiful. Namely, to any polyhedron A, let us attach its “Dehn

invariant” D(A) in V = R ⊗ (R/Q) (the tensor product of Q-vector

spaces). Namely,

D(A) =

a

l(a) ⊗

β(a)

π

,

where a runs over edges of A and l(a),β(a) are the length of a and

the angle at a.

(a) Show that if you cut A into B and C by a straight cut, then

D(A) = D(B) + D(C).

(b) Show that α = arccos(1/3)/π is not a rational number.

Hint: Assume that α = 2m/n, for integers m, n. Deduce that

roots of the equation x +

x−1

= 2/3 are roots of unity of degree n.

Then show that

xk +x−k

has denominator

3k

and get a contradiction.

(c) Using (a) and (b), show that the answer to Hilbert’s question

is negative. (Compute the Dehn invariant of the regular tetrahedron

and the cube.)

2.14. Tensor products and duals of

representations of Lie algebras

Definition 2.14.1. The tensor product of two representations

V, W of a Lie algebra g is the space V ⊗ W with

ρV

⊗W

(x) = ρV (x) ⊗ Id + Id ⊗ρW (x).