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On real dimensions of Lie algebras of holomorphic affine vector fields. (English. Russian original) Zbl 1226.17018
Russ. Math. 51, No. 4, 51-64 (2007); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2007, No. 4, 54-67 (2007).
Introduction: Let $$A$$ be a commutative associative algebra over the field of real numbers $$\mathbb R$$. We assume that $$A$$ is of finite rank and has unity. On an $$A$$-smooth $$n$$-dimensional manifold $$M_n$$, we consider holomorphic linear connections $$\nabla$$. A holomorphic vector field $$X$$ on $$M_n$$ is called affine if the Lie derivative of the linear connection $$\nabla$$ along $$X$$ equals zero. The set $$g(M_n)$$ of affine vector fields admits natural structures of Lie algebras over $$A$$ and $$\mathbb R$$. We denote these Lie algebras by $$(g(M_n))^A$$ and $$(g(M_n))^{\mathbb R}$$, respectively.
Fundamental results in the study of dimensions of Lie algebras $$g(M_n)$$ in the case of $$A = \mathbb R$$ were obtained in [2]. In the case when $$A$$ does not coincide with $$\mathbb R$$, the questions concerning dimensions of Lie algebras $$g(M_n)$$ remained open.
In this paper, the following results are obtained:
1. If the rank of $$A$$ is $$m$$ and the dimension over $$A$$ of $$M_n$$ is $$n$$, then the dimension of the Lie algebra $$(g(M_n))^{\mathbb R}$$ over $$\mathbb R$$ is not greater than $$m(n^2+n)$$.
2. If $$\dim_{\mathbb R}(g(M_n))^{\mathbb R} = m(n^2+n)$$, then the torsion tensor field $$T$$ and the curvature tensor field $$R$$ of $$\nabla$$ equal zero.
3. If the Weyl tensor field $$W$$ of $$\nabla$$ is nonzero, then
$\dim_{\mathbb R}(g(M_n))^{\mathbb R} \leq m(n^2+n)-r_0(3n-5),$
where $$r_0$$ is the singular rank of $$A$$.

##### MSC:
 17B66 Lie algebras of vector fields and related (super) algebras 53C05 Connections, general theory
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##### References:
 [1] V. V. Vishnevskii, A. P. Shirokov, and V. V. Shurygin, Spaces over Algebras (Kazan University, Kazan, 1985). [2] I. P. Yegorov, ”Motions in Affinely Connected Spaces,” in Uchen. Zap. Penzensk. Gos. Ped. Inst. (Kazan University, Kazan, 1965), pp. 3–179. [3] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry (Interscience Publishers, NY, 1963; Nauka, Moscow, 1981). · Zbl 0119.37502 [4] A. Ya. Sultanov, ”On Maximal Dimension of Intransitive Groups of Motions of Affinely Connected Spaces,” in Motions in Generalized Spaces, Mezhvuzovsk. sb. nauchn. tr. (Penza, 2000), pp. 79–90.
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