On the nilpotency of Engel algebras of Lie type (Q2387093)

Let \(G\) be an abelian group. \(L\) is called a \(G\)-graded algebra over a field \(K\) if there exists a finite subset \(P\) of \(G\) such that \(L=\bigoplus _{\alpha \in P}L_\alpha\), where \(L_\alpha\) is a vector space over \(K\) and \(L_\alpha L_\beta \subseteq L_{\alpha +\beta}\), \(\alpha ,\beta ,\alpha +\beta \in G\) and \(L_\gamma = 0\) if \(\gamma \notin P\). A \(G\)-graded algebra \(L\) is called a Lie-type algebra if, for any \(\alpha ,\beta ,\gamma \in P\), there exist \(\lambda ,\mu \in K\), \(\lambda \neq 0\), such that for any homogeneous elements \(e_\alpha (e_\beta e_\gamma)=\lambda (e_\alpha e_\beta)e_\gamma +\mu (e_\alpha e_\gamma )e_\beta\). For \(\lambda =1\), \(\mu =0\), we have an associative algebra and for \(\lambda =1\), \(\mu =-1\), a Lie algebra is obtained. Lie-type algebras include Lie superalgebras, quantum Lie algebras, Witt algebras and color superalgebras. In this paper, the author proves an analog of the Engel theorem in the theory of Lie algebras for Lie-type algebras. In particular, it is proved that a finite-dimensional Lie-type algebra is nilpotent if and only if the operator \(R_x\) is nilpotent for any \(x \in L\). This result is used to study the existence of Cartan subalgebras in anticommutative finite-dimensional Lie-type algebras with ordered grading.