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Publications [#360690] of Curtis W. Porter
Papers Published
 Porter, C; Zelenko, I, Absolute parallelism for 2nondegenerate CR structures via bigraded Tanaka prolongation,
Journal Fur Die Reine Und Angewandte Mathematik, vol. 2021 no. 777
(August, 2021),
pp. 195250 [doi]
(last updated on 2022/08/06)
Abstract: This article is devoted to the local geometry of everywhere 2nondegenerate CR manifolds M of hypersurface type. An absolute parallelism for such structures was recently constructed independently by Isaev and Zaitsev, Medori and Spiro, and Pocchiola in the minimal possible dimension (dimM=5{\dim M=5}), and for dimM=7{\dim M=7} in certain cases by the first author. In the present paper, we develop a bigraded (i.e., ×{\mathbb{Z}\times\mathbb{Z}}graded) analog of Tanaka's prolongation procedure to construct an 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 symbola complex bigraded vector spacecontaining all essential information about the CR structure. Under the additional regularity assumption that the symbol is a Lie algebra, we define a bigraded analog of the Tanaka universal algebraic prolongation, endowed with an antilinear involution, 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 the bigraded universal algebraic prolongation. Moreover, we show that for each regular symbol 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 universal algebraic prolongation of the symbol. In the case of 1dimensional Levi kernel we classify all regular symbols and calculate their bigraded universal algebraic prolongations. In this case, the regular symbols can be subdivided into nilpotent, strongly nonnilpotent, and weakly nonnilpotent. The bigraded universal algebraic prolongation of strongly nonnilpotent regular symbols is isomorphic to the complex orthogonal algebra (m,â ){\mathfrak{so}(m,\mathbb{C})}, where m=12(dimM+5){m=\tfrac{1}{2}(\dim M+5)}. Any real form of this algebraexcept (m){\mathfrak{so}(m)} and (m1,1){\mathfrak{so}(m1,1)}corresponds to the real part of the bigraded universal algebraic prolongation of exactly one strongly nonnilpotent regular CR symbol. However, for a fixed dimM≥7{\dim M\geq 7} the dimension of the bigraded universal algebraic prolongations of all possible regular CR symbols achieves its maximum on one of the nilpotent regular symbols, and this maximal dimension is 14(dimM1)2+7{\frac{1}{4}(\dim M1)^{2}+7}.


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