Open AccessLie algebras of differentiations of linear algebras over a field
Author(s) -
A. Ya. Sultanov,
M.V. Glebova,
O.V. Bolotnikova
Publication year - 2021
Publication title -
differencialʹnaâ geometriâ mnogoobrazij figur
Language(s) - English
Resource type - Journals
eISSN - 2782-3229
pISSN - 0321-4796
DOI - 10.5922/0321-4796-2021-52-12
Subject(s) - mathematics , lie conformal algebra , lie algebra , affine transformation , pure mathematics , rank (graph theory) , algebra over a field , basis (linear algebra) , differential geometry , system of linear equations , universal enveloping algebra , vector field , mathematical analysis , combinatorics , geometry
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.