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On Lie algebras of affine vector fields of ideal realizations of holomorphic linear connections. (English. Russian original) Zbl 1160.53320
Russ. Math. 52, No. 4, 53-58 (2008); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2008, No. 4, 59-65 (2008).
Summary: 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 \(\nabla\) on \(M_n\) over \(\mathbb{A}_m\) (\(m \geq 2\)) is torsion-free and \(R \neq 0\), then the dimension over \(\mathbb R\) of the Lie algebra of all affine vector fields of the space \((M_{mn} ^{\mathbb R}, \nabla ^{\mathbb R})\) is not greater than \((mn)^{2} - 2mn + 5\), where \(m = \text{dim}_{\mathbb R}\mathbb{A}, n = \text{dim}_\mathbb{A} M_n\), and \(\nabla ^{\mathbb R}\) is the real realization of the connection \(\nabla \). Let \(\nabla ^{\mathbb R} =^{1} \nabla \times ^{2} \nabla \) be the real realization of a holomorphic linear connection \(\nabla \) over the algebra of double numbers. If the Weyl tensor \(W = 0\) and the components of the curvature tensor \(^{1} R \neq 0, ^{2} R \neq 0\), then the Lie algebra of infinitesimal affine transformations of the space \((M_{2n} ^{\mathbb R}, \nabla ^{\mathbb R})\) is isomorphic to the direct sum of the Lie algebras of infinitesimal affine transformations of the spaces \((^a M_n, ^a \nabla ) (a = 1, 2)\).

MSC:
53B05 Linear and affine connections
17B66 Lie algebras of vector fields and related (super) algebras
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References:
[1] V. V. Vishnevskii, A. P. Shirokov, and V. V. Shurygin, Spaces over Algebras (Kazan University, Kazan, 1985) [in Russian].
[2] V. V. Vishnevskii, ”On a Property of Analytic Functions over Algebras and its Application to the Study of Complex Structures in Riemannian Spaces,” Uchen. Zap. Kazansk. Univ., N. 126, 5–12 (1966).
[3] I. P. Yegorov, ”Motions in Affinely Connected Spaces,” Uchen. Zap. Penzensk. Gos. Ped. Inst. (Kazan University, Kazan, 1965), P. 3–179.
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