Graded Lie algebras with classical reductive null component
In this work we investigate finite-dimensional graded Lie algebras \(G=\oplus _ iG_ i\) over an algebraically closed field of characteristic \(p>5\) whose null component \(G_ 0\) is classical reductive (the direct sum of ideals each of which is abelian, classical simple, or sl(n), gl(n), \(gl(n)/F_ 1\) with p \(| n)\). In 1970 \textit{V. G. Kac} [Izv. Akad. Nauk SSSR, Ser. Mat. 34, 385-408 (1970; Zbl 0254.17007)] proved that if such an algebra G satisfies 4 additional constraints, then it is a classical Lie algebra with a standard grading, \(gl(m)/F_ 1\) where p \(| m\), or a graded Lie algebra of Cartan type. Kac's theorem, the so-called ``Recognition Theorem'', is an important ingredient in the recent classification of the restricted simple Lie algebras by Block and Wilson. The main result of our paper shows that one of Kac's assumptions, the representation of \(G_ 0\) on \(G_{-1}\) is restricted, is unnecessary. It is expected that this new result will have applicability in the classification of the nonrestricted finite-dimensional simple Lie algebras over algebraically closed fields of prime characteristic.
nonrestricted finite-dimensional simple Lie algebras over algebraically closed fields
scientific article; zbMATH DE number 4057738