On the nilpotency and decomposition of Lie-type algebras.
Let \(G\) be a semigroup acting on a set \(M\). Consider a \(G\)-graded algebra \(B\) over a field \(k\), not necessarily associative, and an \(M\)-graded space \(V\). A Lie-type representation of \(B\) in \(V\) is a linear map \(\rho\) from \(B\) into the endomorphism algebra of \(V\) such that \(\rho(B_\alpha)V_\gamma\subseteq V_{\alpha\gamma}\), and there exists scalars \(\lambda,\mu\in k\) such that \(\rho(ab)v=\lambda\rho(a)\rho(b)v+\mu\rho(b)\rho(a)v\) for homogeneous elements \(a,b\in B\) and \(v\in V\). The aim of the paper is to prove an analog of Engel's theorem for nilpotent Lie algebras. Suppose that \(B\) has finite dimension and \(\rho(x)\) is nilpotent for any homogeneous element \(x\in B\). Then the associative algebra generated by all elements \(\rho(x)\) is nilpotent. If \(V\) has finite dimension then there exists a base in \(V\) in which matrices of all operators \(\rho(x)\), \(x\in B\), are strictly upper triangular. In particular, if \(B\) is a Lie-type left \(B\)-module then \(B\) is nilpotent if and only if each operator of left multiplication on \(B\) is nilpotent. Suppose that \(V=B\) and \(\rho(x)\) is the operator of left multiplication by \(x\). The author introduces an analog of the Killing form. It is shown that if \(G\) is a well-ordered group then a finite dimensional algebra \(B\) is semisimple if and only if the form is non-degenerate. In this case \(B\) is a direct sum of simple homogeneous ideals.
scientific article; zbMATH DE number 5319842