Nijenhuis geometry II: Left-symmetric algebras and linearization problem for Nijenhuis operators |
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Institution: | Faculty of Mechanics and Mathematics, Moscow State University, 119992, Moscow, Russia |
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Abstract: | A field of endomorphisms R is called a Nijenhuis operator if its Nijenhuis torsion vanishes. In this work we study a specific kind of singular points of R called points of scalar type. We show that the tangent space at such points possesses a natural structure of a left-symmetric algebra (also known as pre-Lie or Vinberg-Kozul algebras). Following Weinstein's approach to linearization of Poisson structures, we state the linearisation problem for Nijenhuis operators and give an answer in terms of non-degenerate left-symmetric algebras. In particular, in dimension 2, we give classification of non-degenerate left-symmetric algebras for the smooth category and, with some small gaps, for the analytic one. These two cases, analytic and smooth, differ. We also obtain a complete classification of two-dimensional real left-symmetric algebras, which may be an interesting result on its own. |
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Keywords: | Nijenhuis operator Left-symmetric algebra Linearisation problem |
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