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).
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