1 · The essentials
The result concerns a mathematical object called the rational Grothendieck–Teichmüller Lie algebra. It says that this object can be built from one basic element for each of the labels 3, 5, 7, and so on. These labels are called weights. The basic elements can be combined using a particular operation. The central claim is that no extra constraints on these combinations appear beyond the general rules for this kind of object. The manuscript presents this result as resolving the Deligne–Drinfeld conjecture. The main result has a Lean formalization.
A picture to help (an imperfect image, not to be taken literally)
An imperfect image is a construction kit with one basic piece labeled 3, another labeled 5, another labeled 7, and so on. Every construction would come from those pieces, with no hidden equivalences between assemblies beyond the rules of the kit. This image does not capture the mathematical operation or the precise role of weights.
Why it matters
The result describes the entire algebra through its basic elements and the rules for combining them. It therefore says more than that certain elements exist: they generate the object without additional relations. The abstract also announces the corresponding description after weight completion.
2 · Going further
The problem concerns the Grothendieck–Teichmüller Lie algebra over the rational numbers, equipped with the Ihara bracket. The statement is that it is freely generated by one element in each odd weight . Here, “freely” means that the generators satisfy no additional relations beyond those required by the axioms of a Lie algebra.
The abstract also announces the corresponding isomorphism after weight completion. It thus distinguishes the free description of the algebra from its completed version, which accommodates components of arbitrarily high weight. The coefficient field, bracket, and weights are integral parts of the statement.
The supplied text identifies this statement as the Deligne–Drinfeld conjecture and marks the main result as Lean-formalized. It describes neither the main idea of the argument nor its steps, so an account of the method cannot be extracted from this abstract alone. Only one manuscript is supplied, with no companion manuscript whose role could be explained.
Useful notions
- Rational numbers
- Lie algebras and brackets
- Generators and relations
- Weights and completion
What was known before / what is new
Before
The source identifies the statement as the Deligne–Drinfeld conjecture. It provides neither a history of its formulation nor earlier partial results.
New
The manuscript announces that the rational Grothendieck–Teichmüller Lie algebra, with the Ihara bracket, is freely generated by one element in every odd weight at least three. It also announces the corresponding isomorphism after weight completion. The main result has a Lean formalization.
Limits and assumptions
The statement specifically concerns the rational algebra, the Ihara bracket, and odd weights at least three; the source supplies no generalization beyond this setting. It includes a statement after weight completion. The main result is formalized in Lean, but the supplied text does not detail the scope of that formalization for each assertion, particularly the completion statement. It supplies neither a detailed argument nor any remaining open question.
Mini glossary
- Lie algebra —
- A vector space equipped with an operation called a bracket, subject to the rules defining this type of structure.
- Ihara bracket —
- The bracket operation used in the statement for the rational Grothendieck–Teichmüller Lie algebra.
- Free generation —
- A description in which generators build the entire algebra with no relations beyond those required by the axioms of a Lie algebra.
- Weight —
- A numerical label organizing the components of the algebra, with the announced generators having weights 3, 5, 7, and so on.
- Weight completion —
- Passage to a version of the algebra that accommodates components of arbitrarily high weight.
Keywords : Deligne–Drinfeld conjecture · Grothendieck–Teichmüller algebra · Free generation
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
The Deligne-Drinfeld conjecture
23/09/2026We prove the Deligne–Drinfeld conjecture: the rational Grothendieck–Teichmüller Lie algebra is freely generated by one element in every odd weight at least three, with the corresponding isomorphism after weight completion.