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Differential Geometry of Manifolds

2023 №54(2)

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Sultanov A.Ya. Glebova M.V. Sultanova G.A.

Differentiation of linear algebras with a unit over a field

DOI
10.5922/0321-4796-2023-54-2-5
Pages
54-62
Lie algebra differentiation of linear algebra dimension of Lie algebra linear algebra

Abstract

Linear algebras over a given field arise when studying various problems of algebra, analysis and geometry.  The operation of differentiation, which originated in mathematical analysis, was transferred to the theory of linear algebras over a field, as well as to the theory of rings.

The set of all differentiations of a linear algebra themselves form a linear algebra. This algebra is called the algebra of differentiations. At the same time, this algebra admits the structure of a Lie algebra. If the algebra whose differentiations are considered is finite-dimensional, then its Lie algebra of differentiations will also be finite-dimensional. Therefore, the­re is a natural problem of determining the dimension of the Lie algebras of the differentiations of the linear algebra under consideration or to obtain an estimate from above of the dimension of the algebra of differentiations.

To solve these problems, a system of linear homogeneous equations is obtained, which is satisfied by the components of arbitrary differen­tia­tion. Evaluation of the rank of this system allows us to obtain an estimate from below of the rank of the matrix under consideration.

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