On the nilpotency and decomposition of Lie-type algebras.
type: Item
instance of: scholarly article
title: On the nilpotency and decomposition of Lie-type algebras.
zbMATH Open document ID: 1157.16014
DOI: 10.1134/S0001434607090040
published in: Mathematical Notes
publication date: 2008-09-03T00:00:00Z
review text: 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.
reviewed by: Vyacheslav A. Artamonov
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16N40
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 17A60
zbMATH DE Number: 5319842
zbMATH Keywords: graded algebras
zbMATH Keywords: Engel theorem
zbMATH Keywords: semisimple algebras
zbMATH Keywords: modules of Lie type
zbMATH Keywords: Killing forms
zbMATH Keywords: semisimplicity
author: Nikolai A. Koreshkov
MaRDI profile type: Publication
cites work: Identities of graded algebras
cites work: On the nilpotency of Engel algebras of Lie type
cites work: Q5628302
full work available at URL: s0001434607090040
OpenAlex ID: W2045277975
description: scientific article; zbMATH DE number 5319842