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Cartan subalgebras, weight spaces, and criterion of solvability of finite dimensional Leibniz algebras. (English) Zbl 1128.17001
The paper under review proposes a notion of a Cartan subalgebra in the category of Leibniz algebras, and establishes conditions for the solvability of finite-dimensional Leibniz algebras. This work builds on previous results of the authors [Commun. Algebra 33, No. 1, 159–172 (2005; Zbl 1065.17001)], and draws on analogous theorems for the solvability of finite-dimensional Lie algebras. Recall [J.-L. Loday and T. Pirashvili, Math. Ann. 296, No. 1, 139–158 (1993; Zbl 0821.17022)] that a Leibniz algebra $$L$$ over a ring $$k$$ is a $$k$$-module together with a bilinear bracket $$[\;, \;]: L \times L \to L$$ satisfying the “Leibniz rule”: $[[x, y], z] = [[x, z], y] + [x, [y, z]]$ for all $$x$$, $$y$$, $$z \in L$$. Notions of solvability and nilpotency in the Leibniz case are borrowed from the theory of Lie algebras.
Let $$J$$ be a nilpotent subalgebra of a Leibniz algebra $$L$$. Then the left normalizer of $$J$$ is defined as $l(J) = \{ x \in L\mid[x,J] \subseteq J \}.$ Since the bracket $$[\;, \;]$$ is not necessarily skew-commutative, the left normalizer does not, in general, coincide with the right normalizer. A subalgebra $$J$$ is then called a Cartan subalgebra if $$J$$ is nilpotent and $$J = l(J)$$. Let $$R_x : L \to L$$ be the linear transformation of $$L$$ given by $R_x(a) = [a,x], \quad a \in L,$ and define $R(J) = \{ R_x\mid x \in J \}.$ Conditions for $$J$$ to be a Cartan subalgebra are given in terms of $$R(J)$$ and $$R_x$$ for regular elements $$x \in L$$. Proven is that a nilpotent subalgebra $$J$$ of $$L$$ is a Cartan subalgebra if and only if $$J$$ equals a certain null-space in the decomposition of $$L$$ with respect to $$R(J)$$. Additionally, the authors prove that for a Leibniz algebra $$L$$ over an algebraically closed field of characteristic zero, $$L$$ is solvable if and only if tr$$(R_a \, R_a) = 0$$ for all $$a \in [L,L]:= L^2$$.

##### MSC:
 17A32 Leibniz algebras 17B30 Solvable, nilpotent (super)algebras 17A60 Structure theory for nonassociative algebras
##### Keywords:
Leibniz algebras; Cartan subalgebras; weight spaces
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