On lie algebras of affine vector fields of ideal realizations of holomorphic linear connections |
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Authors: | M. V. Morgun A. Ya. Sultanov |
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Affiliation: | (1) Penza State Pedagogical University, ul. Lermontova 37, Penza, 440026, Russia |
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Abstract: | We study the properties of real realizations of holomorphic linear connections over associative commutative algebras (mathbb{A}) m with unity. The following statements are proved.If a holomorphic linear connection ? on M n over (mathbb{A}) m (m ≥ 2) is torsion-free and R ≠ 0, then the dimension over ? of the Lie algebra of all affine vector fields of the space (M mn ? , ??) is no greater than (mn)2 ? 2mn + 5, where m = dim? (mathbb{A}), (n = dim_mathbb{A} ) M n , and ?? is the real realization of the connection ?.Let ?? =1 ? ×2 ? be the real realization of a holomorphic linear connection ? over the algebra of double numbers. If the Weyl tensor W = 0 and the components of the curvature tensor 1 R ≠ 0, 2 R ≠ 0, then the Lie algebra of infinitesimal affine transformations of the space (M 2n ? , ??) is isomorphic to the direct sum of the Lie algebras of infinitesimal affine transformations of the spaces ( a M n , a ?) (a = 1, 2). |
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Keywords: | holomorphic linear connection real realization Lie algebra of infinitesimal affine transformations |
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