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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