nLab holonomy Lie algebra

Given a connection on a principal $G$-bundle $P\to M$ the holonomy Lie algebra at a point $p$ in a total space principal bundle $P$ is the tangent Lie algebra of the holonomy group at $p$ of a principal $G$-bundle. By the Ambrose-Singer theorem the holonomy Lie algebra is spanned as a vector subspace of the Lie algebra $\mathfrak{g}$ of $G$ spanned by all elements $\Omega_q(X,Y)$ where $q$ is a point in $P$ connected to $q$ by some horizontal path and $X,Y\in H_q P\subset T_q P$ are horizontal vectors at $q$.

Literature

• M. M. Postnikov, Lectures on differential geometry IV

There are particular cases related to arrangements of hyperplanes like Drinfeld-Kohno Lie algebra and its generalizations.

• M. V. Feigin, A. P. Veselov, $\vee$-systems, holonomy Lie algebras, and logarithmic vector fields, Intern. Math. Res. Notices 2018, No. 7, pp. 2070-2098 doi

For higher categorical analogue see

• Camilo Arias Abad, Florian Schätz, Holonomies for connections with values in $L_\infty$-algebras, Homology, Homotopy and Applications 16:1 (2014) 89-118

category: geometry