Absolute parallelism for 2-nondegenerate CR structures via bigraded Tanaka prolongation
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An absolute parallelism for $2$-nondegenerate CR manifolds $M$ of hypersurface type was recently constructed independently by Isaev-Zaitsev, Medori-Spiro, and Pocchiola in the minimal possible dimension ($dim M=5$), and for $dim M=7$ in certain cases by the first author. We develop a bigraded analog of Tanaka's prolongation procedure to construct a canonical absolute parallelism for these CR structures in arbitrary (odd) dimension with Levi kernel of arbitrary admissible dimension. We introduce the notion of a bigraded Tanaka symbol. Under regularity assumption that the symbol is a Lie algebra, we define a bigraded analog of the Tanaka universal algebraic prolongation and prove that for any CR structure with a given regular symbol there exists a canonical absolute parallelism on a bundle whose dimension is that of this bigraded prolongation. We show that there is a unique (up to local equivalence) such CR structure whose algebra of infinitesimal symmetries has maximal possible dimension, and the latter algebra is isomorphic to the real part of the bigraded prolongation of the symbol. In the case of $1$-dimensional Levi kernel we classify all regular symbols and calculate their bigraded prolongations. In this case the regular symbols can be subdivided into nilpotent, strongly non-nilpotent and weakly non-nilpotent. The bigraded prolongation of strongly non-nilpotent symbols is isomorphic to $mathfrak{so}left(m,mathbb C ight)$ where $m= frac{1}{2}(dim M+5)$. Any real form of this algebra, except $mathfrak{so}left(m ight)$ and $mathfrak{so}left(m-1,1 ight)$, corresponds to the real part of the bigraded prolongation of exactly one strongly non-nilpotent symbol. However, for a fixed $dim Mgeq 7$ the dimension of the bigraded prolongations achieves its maximum on one of the nilpotent regular symbols, and this maximal dimension is equal to $ frac{1}{4}(dim M-1)^2+7$.