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Grothendieck-Teichmüller Lie Algebra

The Grothendieck-Teichmüller Lie algebra is the Lie algebra of infinitesimal symmetries of associativity and braiding, expressed by antisymmetry, a three-term relation, and a pentagon relation in free Lie algebras. In the graded convention, its elements are Lie polynomials psi(x,y) satisfying

psi(x,y)+psi(y,x)=0,
(1)
psi(x,y)+psi(y,-x-y)+psi(-x-y,x)=0,
(2)

and the pentagon identity

psi(t_(12),t_(23)+t_(24))+psi(t_(13)+t_(23),t_(34))=psi(t_(23),t_(34))+psi(t_(12)+t_(13),t_(24)+t_(34))+psi(t_(12),t_(23)).
(3)

The last identity is imposed in the infinitesimal pure braid Lie algebra on four labels. Its generators t_(ij)=t_(ji) satisfy [t_(ij),t_(kl)]=0 for distinct indices and [t_(ij),t_(ik)+t_(jk)]=0 for distinct i, j, and k.

For a Lie polynomial psi, let D_psi be the derivation with D_psi(x)=0 and D_psi(y)=[y,psi]. The Ihara bracket in this convention is

{psi,phi}=D_psi(phi)-D_phi(psi)+[psi,phi].
(4)

Completion by total degree gives the completed version. The Deligne-Drinfeld conjecture predicts its free odd-weight generators.

Replacing the even free generators by odd generators gives the odd Grothendieck-Teichmüller Lie algebra. Naef and Willwacher (2026) prove that this analog is free on a single odd generator of weight 3.

See also

Deligne-Drinfeld Conjecture, Derivation, Free Lie Algebra, Lie Algebra, Lie Bracket

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References

Alm, J. "The Grothendieck-Teichmueller Lie Algebra and Brown's Dihedral Moduli Spaces." 17 May 2018. https://arxiv.org/abs/1805.06684.Naef, F. and Willwacher, T. "An Odd Version of the Grothendieck-Teichmüller Lie Algebra and the Odd Deligne-Drinfeld Conjecture." 30 Oct 2026. https://arxiv.org/abs/2610.41766v1.

Cite this as:

Weisstein, Eric W. "Grothendieck-Teichmüller Lie Algebra." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Grothendieck-TeichmuellerLieAlgebra.html

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