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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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##### References:
 [1] Y. Bahturin and M. Zaicev, ”Identities of graded algebras,” J. Algebra 205(1), 1–12 (1998). · Zbl 0920.16011 · doi:10.1006/jabr.1997.7017 [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)]. · doi:10.4213/sm437 [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].
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