The Grothendieck–Teichmüller Lie algebra and the Deligne–Drinfeld conjecture

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The Grothendieck–Teichmüller Lie algebra and the Deligne–Drinfeld conjecture

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Recursos

Pick any weight n and work out, with paper and pencil, how many solutions the pentagon, hexagon and antisymmetry equations can have there. The characteristic-two argument behind the Deligne–Drinfeld conjecture turns that count into a short procedure, and this class follows it one step at a time.

The squeeze

The rational solutions form a graded space W ⊂ Lie_ℚ⟨x, y⟩. Brown's motivic work gives a free Lie algebra of solutions from below, with one generator in each odd weight ≥ 3. The new part is the upper bound dim W_n ≤ d_n, where d_n is the weight-n dimension of that same free Lie algebra. Once both bounds hold, nothing else fits [1:05].

The chain, step by step

Reduce mod 2 without losing dimension. The saturated ℤ₍₂₎-lattice of solutions has a reduction of the same dimension [1:40].
New letters: A = x², C = [x, y], B = y, with weights 2, 2 and 1. Over 𝔽₂, [x,[x,y]] = [A, B]. The leading words xx, xy, y form a prefix code, so every Lie polynomial of weight > 1 lives in Lie⟨A, C, B⟩.
Filter by the number of B's, r, so that n = 2m + r. Applied to the leading part, the pentagon degenerates to a relation in the level-two cyclotomic arrangement algebra [2:37].
Set A = 0. Deletion operators, together with a special identity proved over ℚ, show that this "shadow" is injective [3:04].
Shift symmetry + Vandermonde. The projected part f is unchanged by B ↦ B + s(C). A Vandermonde determinant ∏(t_b − t_a) ≠ 0 then forces f into the free Lie algebra on g_k = ad_C^k B, which has weight 2k + 1 [3:47].

Counting a weight

To count a weight, list the lengths r with the same parity as n, then count the Lie words in the g_k of weight n and length r. For n = 8, only r = 2 works, giving [g₁, g₂], so d₈ = 1 [4:59]. For n = 11, you get g₅ and [g₁,[g₁,g₂]], so d₁₁ = 2. Length 2 is impossible there, because two odd weights always add up to an even number [5:37]. The other half of the proof, which builds solutions that reach the bound through holonomy, is in Sections 6–8 of the paper [6:14].

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