Unipotent automorphisms of modular group algebras and the homology of the Lazard Lie algebra
Abstract
Let p be an odd prime and H a finite p-group. Hertweck and Soriano parametrize the central Frattini extensions of H by subspaces of a relation module V, matching isomorphism classes of extensions with orbits of Out(H) and isomorphism classes of the associated small group rings with orbits of Out(𝔽p H). The discrepancy between the two is measured by the group U(H)≤ GL(V) of displacements induced by the normalized unipotent automorphisms of 𝔽p H; it is the mechanism behind the counterexamples to the modular isomorphism problem in characteristic $2$. We prove that for H of exponent p and class at most p−2 every such automorphism acts on V exactly as an automorphism of H does, and that U(H)=expΩ(Der>0(l(H))), where l(H) is the Lazard Lie algebra of H, Der>0 its derivations raising the lower central filtration, and Ω the natural action of derivations on H2(l(H)). Hence U(H)=1 precisely when every derivation of positive degree acts trivially on H2(l(H)). We compute U(H) for all groups of class $2$ and exponent p, obtaining p2n(n−1) for the Heisenberg groups Heis(pn).
Related
- Modular Zhu algebra theory and Virasoro vertex algebras
- Aperiodic automorphisms of the Razak--Jacelon algebra
- Automorphisms of general modular curves
- Transposed Poisson structure on the Witt-type algebra 𝒲(a,−1): Derivations, Automorphisms, and Rota--Baxter operators
- The Quillen and sharbly Hopf algebra structures on Steinberg homology coincide
BibTeX
@misc{tuzzolo2026unipotent,
title = {{Unipotent automorphisms of modular group algebras and the homology of the Lazard Lie algebra}},
author = {Diego Tuzzolo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06909},
eprint = {2610.06909},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06909},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Unipotent automorphisms of modular group algebras and the homology of the Lazard Lie algebra
AU - Diego Tuzzolo
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.06909
UR - https://arxiv.org/abs/2610.06909
AB - Let p be an odd prime and H a finite p-group. Hertweck and Soriano parametrize the central Frattini extensions of H by subspaces of a relation module V, matching isomorphism classes of extensions with orbits of Out(H) and isomorphism classes of the associated small group rings with orbits of Out(𝔽_p H). The discrepancy between the two is measured by the group U(H)≤ GL(V) of displacements induced by the normalized unipotent automorphisms of 𝔽_p H; it is the mechanism behind the counterexamples to the modular isomorphism problem in characteristic $2$. We prove that for H of exponent p and class at most p−2 every such automorphism acts on V exactly as an automorphism of H does, and that U(H)=expΩ(Der_>0(l(H))), where l(H) is the Lazard Lie algebra of H, Der_>0 its derivations raising the lower central filtration, and Ω the natural action of derivations on H_2(l(H)). Hence U(H)=1 precisely when every derivation of positive degree acts trivially on H_2(l(H)). We compute U(H) for all groups of class $2$ and exponent p, obtaining p^2n(n−1) for the Heisenberg groups Heis(p^n).
ER -