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 \(A\).

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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\textit{A. Ya. Sultanov}, Russ. Math. 51, No. 4, 51--64 (2007; Zbl 1226.17018); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2007, No. 4, 54--67 (2007)

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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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