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Odd Grothendieck-Teichmüller Lie Algebra

The odd Grothendieck-Teichmüller Lie algebra is a Lie superalgebra over Q defined by odd symmetry relations (Naef and Willwacher 2026, Definition 1.1). Let L(x,y) be the free Lie superalgebra on odd generators x,y of weight 1. For each n, let t_n be generated by odd elements t_(ij) with t_(ji)=-t_(ij), with [t_(ij),t_(kl)]=0 for distinct i,j,k,l and [t_(ij),t_(ik)+t_(jk)]=0 for distinct i,j,k. The odd Grothendieck-Teichmüller Lie algebra consists of the psi(x,y) in L(x,y) satisfying

psi(x,y)+psi(-y,-x)=0,
(1)
psi(t_(12),t_(23))+psi(t_(23),t_(31))+psi(t_(31),t_(12))=0,
(2)

and

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 two identities are imposed in t_3 and t_4, respectively. For a homogeneous f, let partial_f be the graded derivation of parity |f| with partial_f(x)=0 and partial_f(y)=-[f,y]. The graded Ihara bracket is

{f,g}=[f,g]+partial_f(g)-(-1)^(|f||g|)partial_g(f).
(4)

Parity is weight modulo 2.

Naef and Willwacher (2026, Theorem 1.3) prove that it is the free graded Lie superalgebra on one odd generator sigma_3 of weight 3. It has basis sigma_3,{sigma_3,sigma_3}, of weights 3 and 6. Every other weight component vanishes. This is the odd analog of the Deligne-Drinfeld conjecture, rather than a resolution of its classical even-generator form. Naef and Willwacher (2026) report substantive AI assistance with proof ideas and tools.

See also

Deligne-Drinfeld Conjecture, Lie Superalgebra

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References

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. "Odd Grothendieck-Teichmüller Lie Algebra." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OddGrothendieck-TeichmuellerLieAlgebra.html

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