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On the nilpotency and decomposition of Lie-type algebras. (English. Russian original) Zbl 1157.16014
Math. Notes 82, No. 3, 321-331 (2007); translation from Mat. Zametki 82, No. 3, 361-372 (2007).
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.
MSC:
16W50 Graded rings and modules (associative rings and algebras)
16N40 Nil and nilpotent radicals, sets, ideals, associative rings
17B70 Graded Lie (super)algebras
17A60 Structure theory for nonassociative algebras
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