On the centre of crossed modules of Lie algebras
Mariam Pirashvili
TL;DR
The paper develops a centre theory for crossed modules of Lie algebras by constructing a centre ${\bf Z}_*(L_*)=(L_1\xrightarrow{\delta}{\bf Z}_0(L_*))$, where ${\bf Z}_0(L_*)$ consists of pairs $(x,\xi)$ constrained by compatibility conditions with $\partial$ and the action. It shows that ${\bf Z}_*(L_*)$ is a braided crossed module and that the induced cofibre $L_*//{\bf Z}_*(L_*)$ encodes a homotopical counterpart to the centre, with explicit descriptions of $\pi_1$ and $\pi_0$ of the centre in terms of Lie algebra cohomology: $\pi_1({\bf Z}_*(L_*))\cong H^0(\pi_0(L_*),\pi_1(L_*))$ and an exact sequence $0\to H^1(\pi_0(L_*),\pi_1(L_*))\to \pi_0({\bf Z}_*(L_*))\to Z_{\pi_1(L_*)}(\pi_0(L_*))\to H^2(\pi_0(L_*),\pi_1(L_*))$. The work further relates the centre to Guin's nonabelian Lie algebra cohomology, embedding ${\bf Z}_0(L_*)$ into ${\sf Der}_{L_0}(L_0,L_1)$ and interpreting $H^0$ and $H^1$ in terms of derivations and cocycles, thereby linking centre theory, homotopy, and nonabelian cohomology in the Lie algebra setting.
Abstract
This paper studies the relationship between crossed modules of Lie algebras and their centres. We show that any crossed module \(\partial : L_1\to L_0\) of Lie algebras fits in an exact sequence involving cohomology of the homotopy Lie algebras \(π_0(L_*)\) and \(π_1(L_*)\).