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

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 ℝ of the Lie algebra of all affine vector fields of the space (M ℝ , ∇ℝ) is no greater than (mn)2 − 2mn + 5, where m = dimℝ $$ \mathbb{A} $$ , $$ n = dim_\mathbb{A} $$ M n , and ∇ℝ is the real realization of the connection ∇. Let ∇ℝ =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 2 ℝ , ∇ℝ) is isomorphic to the direct sum of the Lie algebras of infinitesimal affine transformations of the spaces ( a M n , a ∇) (a = 1, 2).