On contact and finitely Levi-nondegenerate CR algebras

We study CR-manifolds of arbitrary CR codimension, mainly focusing on Levi and contact-nondegeneracy and depth. We investigate these and other invariants in the locally homogeneous case, developing a comprehensive theory which establishes correspondences with related properties of the associated CR-algebras and, in the parabolic case, with the combinatorics of their cross-marked painted root diagrams.

💡 Research Summary

The paper develops a comprehensive framework for studying three fundamental invariants of CR manifolds of arbitrary CR‑codimension: Levi‑nondegeneracy, contact‑nondegeneracy, and depth. After recalling the classical definitions of a CR structure, the authors introduce the Levi order (k_p(M)) as the minimal number of iterated Lie brackets of a (1,0) vector field with anti‑holomorphic fields needed to leave the complexified contact distribution, and the contact order (k_c(p)) as the analogous minimal number for real vector fields within the real contact distribution. They prove that (k_c(p)\le k_p(p)), with equality at order 1 corresponding to strict Levi‑nondegeneracy.

To handle locally homogeneous CR manifolds, the authors adopt the notion of a CR‑algebra ((\mathfrak g^\sigma,\mathfrak q)), where (\mathfrak g^\sigma) is the real Lie algebra of a symmetry group (G^\sigma) and (\mathfrak q\subset\mathfrak g) (the complexification of (\mathfrak g^\sigma)) is a complex Lie subalgebra encoding the (0,1) distribution. The integrability condition translates into (