Witt algebra and the curvature of the Heisenberg group (Q2844462)

The notion of curvature algebra of a Finsler manifold was introduced by the authors in [Houston J. Math. 38, No. 1, 77--92 (2012; Zbl 1238.53012)] and it was proved that this algebra is tangent to the holonomy group. This property was used for the proof that the holonomy groups of the Finsler manifolds of constant non-zero curvature cannot be compact Lie groups, if the dimension of the manifold is greater than 2. The aim of this paper is to determine explicitly the algebraic structure of the curvature algebra of the 3-dimensional Heisenberg group with left invariant cubic metric. It is shown that this curvature algebra is an infinite-dimensional graded Lie subalgebra of the generalized Witt algebra of homogeneous vector fields generated by 3-elements.