On derivations of linear algebras of a special type

• Published in: Дифференциальная геометрия многообразий фигур
• Main Authors: A. Ya. Sultanov, O. A. Monakhova, O. V. Bolotnikova
• Format: Article
• Language: English
• Published: Immanuel Kant Baltic Federal University 2024-01-01
• Subjects: linear algebra over a field; linear form; differentiation of linear algebra; lie algebra of differentiation; lie algebra of matrices; unit of linear algebra
• Online Access: https://journals.kantiana.ru/geometry/15698/79878/
• ISSN: 0321-4796; 2782-3229

In this work, Lie algebras of differentiation of linear algebra, the op­eration of multiplication in which is defined using a linear form and two fixed elements of the main field are studied. In the first part of the work, a definition of differentiation of linear algebra is given, a system of linear homogeneous equations is obtained, which is satisfied by the components of arbitrary differentiation. An embedding of the Lie algebra of differenti­ations into the Lie algebra of square matrices of order n over the field P is constructed. This made it possible to give an upper bound for the dimen­sion of the Lie algebra of derivations. It has been proven that the dimen­sion of the algebra of differentiation of the algebras under study is equal to n2 – n, where n is the dimension of the algebra. Next we give a result on the maximum dimension of the Lie algebra of derivations of a linear alge­bra with identity. Based on the above facts, it is proven that the algebras under study cannot have a unit.