Classification of leibniz algebras over finite fields

Classical methods to obtain classifications are essentially to solve a system of equations given by the identities of specified classes of algebras. Since so far there are no research results dealing with representing Leibniz algebras over a finite field, it is desirable to have such lists up to isomorphism over finite fields. We apply the structure constants and some algebraic invariants to obtain complete lists of two-and three-dimensional Leibniz algebras over finite fields. Here an algorithm is also given to classify three-dimensional algebras over any field. Then we apply it to threedimensional Leibniz algebras over some finite fields. The description of the group of automorphisms of low-dimensional Leibniz algebras were also given. The main idea of the annihilator extension is to transfer the “base change” action to an action of the automorphism group of the algebras of smaller dimension on cocycles constructed by the extension. This method has been used earlier to classify certain classes of algebras. We review and extend theoretical background of the method for Leibniz algebras then apply it in our research. Results on automorphism group of threedimensional non-Lie Leibniz algebras obtain in the first part are used to classify four-dimensional non-Lie Leibniz algebras over Zp where p = 3; 5, using the analogue of the Skjelbred and Sund method. The distribution of algebras into equivalent classes is usually done according to the concept of isomorphism. However, such a distribution can also be done into isotopism classes. This relation is weaker than isomorphism relation. Finally, we describe the isotopism classes of low-dimensional Leibniz algebras over finite fields.

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