On Lie algebras of affine vector fields of real realizations of holomorphic linear connections

A. Ya. Sultanov ; M. V. Morgun

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 4 (2008), pp. 59-65.

Résumé

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 $\mathbb A_m$ $(m\ge2)$ is torsion-free and $R\ne0$, 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 no greater than $(mn)^2-2mn+5$, where $m=\dim_{\mathbb R}\mathbb A$, $n=\dim_{\mathbb A}M_n$ 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 $^1R\ne0$, $^2R\ne0$, 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 $(^aM_n,\,^a\nabla)$ $(a=1,2)$..

Détail

Keywords: holomorphic linear connection, real realization
Mots-clés : Lie algebra of infinitesimal affine transformations.

Bibliographie

[1] Vishnevskii V. V., Shirokov A. P., Shurygin V. V., Prostranstva nad algebrami, Izd-vo Kazansk. un-ta, Kazan, 1984, 264 pp. | MR

[2] Vishnevskii V. V., “Ob odnom svoistve analiticheskikh funktsii nad algebrami i ego prilozhenie k izucheniyu kompleksnykh struktur v rimanovykh prostranstvakh”, Uchen. zap. Kazansk. un-ta, 126:1 (1966), 5–12 | MR | Zbl

[3] Egorov I. P., “Dvizheniya v prostranstvakh affinnoi svyaznosti”, Dvizheniya v prostranstvakh affin. svyaznosti, Izd-vo Kazansk. un-ta, Kazan, 1965, 5–179