nLab holonomy Lie algebra


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


  • 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

category: geometry

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