On lie algebras of affine vector fields of ideal realizations of holomorphic linear connections

摘要

We study the properties of real realizations of holomorphic linear connections over associative commutative algebras m with unity. The following statements are proved. If a holomorphic linear connection on M n over 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, M n , and is the real realization of the connection. Let=1×2be 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).