Directed pseudo-graphs and lie algebras over finite fields

Isotopism and Isomorphism Classes of Certain Lie Algebras over Finite Fields

“…The family F p n was introduced by Boza et al [3] for p = 2, where each algebra of the family was uniquely identified with a directed pseudo-graph and their distribution into isomorphism classes was then explicitly obtained for n ≤ 5, by considering the properties of such graphs. More recently, the case p = 3 and n < 5 was similarly analyzed in [4,9]. The known isomorphism classes are shown in Table 1, where the representative elements have been chosen to agree with the results of the current paper.…”

“…If there exists i 0 < n such that f ni0 = 0, then (i) implies that there exists a number α k ∈ F p such that f ki = α k f ni for all i < n. If i 0 > m, then (4) implies that f kn = α k f nn and hence F is non-singular, which is a contradiction. Therefore Let M T = (m ij ) be the matrix of order (n − 1) × n such that m ij = t ij if j < n; otherwise 0.…”

“…, n − 1}. Now, if i 0 ≤ m, then f ki = α k f ni = 0 for all k ≤ m and i > m.Finally, if f ni = 0 for all i ≤ m, then f nn = 0 because F is non-singular and thus,(4) implies again that f ki = 0 for all k ≤ m and i > m.…”