On derivations of linear algebras of a special type
Abstract
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.
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