On the solvability of the Lie algebra $\mathrm{HH}^1(B)$ for blocks of finite groups
Markus Linckelmann, Jialin Wang
TL;DR
The paper develops criteria for the solvability of the Lie algebra $\mathrm{HH}^1(B)$ of blocks $B$ of group algebras, tying solvability to the action of the inertial quotient $E$ on a defect group $P$ via the ${\mathbb F_p}E$-module structure of $P/\Phi(P)$. It leverages Puig’s Morita-type stable equivalence $B \simeq k_\alpha(P\rtimes E)$ and analyzes $\mathrm{HH}^1$ through twisted group algebras, the K"unneth formula, and $E$-stable derivations on $kP$; a key outcome is that multiplicity-free action of $E$ on $P/\Phi(P)$ yields solvability, with an odd $p$ converse. The results provide a clear, computable criterion for solvability in many block situations, including abelian defect groups and Frobenius-type blocks, and they connect the global block structure to local fusion-system invariants. Together with explicit examples, the work clarifies how $\mathrm{HH}^1(B)$ can reflect simple, solvable, or non-solvable Lie algebra structures across block families, aiding both theoretical understanding and potential computational checks.
Abstract
We give some criteria for the Lie algebra $\mathrm{HH}^1(B)$ to be solvable, where $B$ is a $p$-block of a finite group algebra, in terms of the action of an inertial quotient of $B$ on a defect group of $B$.