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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
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