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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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