Short··0:21·Number theory
Deligne–Drinfeld Conjecture: One Generator per Odd Weight
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About this video
The Deligne–Drinfeld conjecture predicts the exact shape of the Grothendieck–Teichmüller Lie algebra grt, a symmetry algebra tied to braids, multiple zeta values and the absolute Galois group of the rationals: it should be freely generated by one element in each odd weight 3, 5, 7, 9, and so on. Brown proved in 2012 that such a free Lie algebra sits inside grt; OpenAI claims a proof that it is all of grt, with the corresponding isomorphism after weight completion (an OpenAI preprint, not yet peer reviewed). The picture shows the real basis of that free Lie algebra up to weight 20: each rod is one independent iterated bracket (a Lyndon word), cut into pieces for the generators it uses, sitting on the step of the stair equal to its weight; the counts 1, 0, 1, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 7, 8, 11, 13 are the dimensions in weights 3 to 20.
Paper (OpenAI, catalogue result 008):
- The Deligne-Drinfeld conjecture: github.com/openai/math/blob/main/preprints/The-Deligne-Drinfe…
The animation illustrates the result; it is not a proof.
Gradient Damage: mathematics you can see, equations you can hear.
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Every result explained: gradientdamage.com/p/008
About the problem
Number theory·OpenAI Math·Result 008
The Deligne–Drinfeld conjecture
Proves that the rational Grothendieck–Teichmüller Lie algebra, with the Ihara bracket, is freely generated by one element in each odd weight 3, 5, 7, …, resolving the Deligne–Drinfeld conjecture. (Lean)
Topics
Deligne-Drinfeld conjectureDeligne Drinfeld conjectureGrothendieck-Teichmüller Lie algebraGrothendieck-Teichmüller groupgrtfree Lie algebraLie algebraLie bracketLyndon wordsmultiple zeta valuesmotivic Galois groupIhara bracketFrancis Brownnumber theoryOpenAIOpenAI mathAI mathematicsAI proofmathsmathematicsmath shorts3D animationGrothendieck-Teichmuller Lie algebraGrothendieck-Teichmuller group
