Lie algebras of differentiations of linear algebras over a field

Abstract

In this paper, we study a system of linear equations that define the Lie algebra of differentiations DerA of an arbitrary finite-dimensional linear algebra over a field. A system of equations is obtained, which is satisfied by the components of an arbitrary differentiation with respect to a fixed basis of algebra A. This system is a system of linear homogeneous equa­tions. The law of transformation of the matrix of this system is proved. The invariance of the rank of the matrix of this system in the transition to a new basis in algebra is proved. Next, we consider the possibility of ap­plying the obtained results in differential geometry when estimating the dimensions of groups of affine transformations from above. As an exam­ple, the method of I. P. Egorov is given for studying the dimensions of Lie algebras of affine vector fields on smooth manifolds equipped with linear connections having non-zero torsion tensor fields.