×

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Y. Bahturin and M. Zaicev, ”Identities of graded algebras,” J. Algebra 205(1), 1–12 (1998). · Zbl 0920.16011
[2] Yu. A. Bakhturin, M. V. Zaitsev, and S. K. Segal, ”G-identities of nonassociative algebras,” Mat. Sb. 190(11), 3–14 (1999) [Sb. Math. 190 (11–12), 1559–1570 (1999)].
[3] N. A. Koreshkov, ”On the nilpotency of Lie-type Engel algebras,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 3, 36–40 (2004) [Russian Math. (Iz. VUZ) 48 (3), 32–36 (2004)]. · Zbl 1100.17011
[4] K. A. Zevlakov [Zhevlakov], ”Nil-radical of a Malcev algebra,” Algebra i Logika Sem. [Algebra i Logica] 4(5), 67–78 (1965).
[5] V. T. Filippov, ”The Engel algebras of Malcev,” Algebra i Logika 15(1), 89–109 (1976).
[6] E. N. Kuz’min, ”Anticommutative algebras satisfying Engel’s condition,” Sibirsk. Mat. Z. 8(5), 1026–1034 (1967).
[7] I. Kaplansky, Lie Algebras and Locally Compact Groups (The University of Chicago Press, Chicago, Ill.-London, 1971; Mir, Moscow, 1974). · Zbl 0223.17001
[8] N. G. Chebotarev, Introduction to the Theory of Algebras (Gostekhizdat, Moscow-Leningrad, 1949) [in Russian].
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.