OAI.DeligneDrinfeld.main

The theorem states that the defined proposition MainStatement holds. Let L be the free Lie algebra over ℚ on two generators x and y, graded by word length (Ln n is the span of brackets of generators of total length n), and let eval(a,b) be the Lie homomorphism from L sending x to a and y to b. Let T4 be the free Lie algebra on the six chords t_ij (0≤i<j≤3, with t_ji = t_ij and t_ii = 0) modulo the ideal generated by [t_ij,t_kl] for four distinct indices and [t_ij, t_ik + t_jk] for three distinct indices. W is the ℚ-subspace of p in L satisfying three equations: p + eval(y,x)p = 0; p + eval(y,−x−y)p + eval(−x−y,x)p = 0; and a pentagon-type relation in T4, eval(t01, t12+t13)p + eval(t02+t12, t23)p = eval(t12,t23)p + eval(t01+t02, t13+t23)p + eval(t01,t12)p. The Ihara bracket is ihara(p,q) = D(p)(q) − D(q)(p) + [p,q], where D(p) is the derivation of L determined by x ↦ 0 and y ↦ [y,p], extracted from a Lie homomorphism into a tangent Lie algebra. Let OddFree be the free Lie algebra over ℚ on generators indexed by k ∈ ℕ, with generator k of weight 2k+3 and weight pieces Fn n. The claim is that there exist a ℚ-linear isomorphism e from OddFree onto W and a proof that e is graded (e a lies in Ln n exactly when a lies in Fn n) such that: e carries the Lie bracket to the Ihara bracket, e[a,b] = ihara(e a, e b); the induced componentwise map between the products of the graded pieces Fn n and W∩Ln n has a two-sided inverse and is continuous in both directions (product topologies, with L and OddFree taken discrete); and for all sequences a, b in the completion and every n, e applied to the sum over i from 0 to n of [a_i, b_{n−i}] equals the sum over i from 0 to n of ihara of the completed images of a and b at degrees i and n−i. The theorem is stated with an admitted proof.