Finite-Dimensional Simple Lie Algebras with a Nonsingular Derivation

Gradings of non-graded Hamiltonian Lie algebras

“…Note that the Hamiltonian algebras in two variables can also be considered as belonging to the Cartan series of special algebras (and thus be denoted by 5(2 : n; &>,)). A rather condensed description of all four classes of Lie algebras of Cartan type, but complete with all the relevant references, can be found in [7], to which we also conform our notation. (See Remark 3.1 concerning our notation.)…”

“…This algebra has a non-degenerate associative form k (see [11,Theorem 7]), defined by the same formula given above for H{2 : n;co 0 ) with, in addition, k(xy, xy) -1 and k(-, •) = 0 in all remaining cases. Again from [7] (or the original source [41,Theorem 3.2]), all outer derivations of H(2 : n;<w 2 ) are lifted from part of those of its associated graded algebra, which is H(2 : n\co 0 ) © {xy). Specifically, a basis for the space of outer derivations of H(2 : n;o> 2 ) is given by (adjt) p ' forO < r < n 2 and (ady) p ' forO < 5 < n,.…”

“…Benkart, Kostrikin and Kuznetsov proved in [7,Theorem 4.9] (using the classification of the modular simple Lie algebras of characteristic p > 1 completed in [56]) terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700010983 that the only simple Lie algebras over an algebraically closed field of characteristic p > 7 which admit a nonsingular derivation agreeing with a Z/NZ-grading with onedimensional components (in the terminology introduced in Definition 5.1) are of type H(m : n;cc>2).…”

“…The Block algebras used in [20] are actually isomorphic with algebras of Albert and Frank (being simple Block algebras with G = G o , see Section 3, and according to known results, for example [13,Lemma 1.8.3]), but were presented there in a different basis. As we have mentioned earlier, the original motivation for the present paper was finding an explicit identification of those algebras with Hamiltonian algebras H{2 : n; <o 2 ) and describing a corresponding cyclic grading [7] Gradings of non-graded Hamiltonian Lie algebras 405 of the latter. We realize these two tasks in Sections 8 and 7, respectively.…”

“…Benkart, Kostrikin and Kuznetsov proved in [7] and [39] that the only finite-dimensional simple Lie algebras with this property, over an algebraically closed field of characteristic p > 7, are the Hamiltonian algebras H(2: n;a>2). The proof rests on the classification of simple modular Lie algebras, which causes the restriction on p, but now a classification-free proof of this result for p > 2 is a consequence of [24].…”