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**Associative and Lie algebras of quotients.**
*(English)*
Zbl 1151.17010

In 2001 C. Martínez gave a necessary and sufficient Ore-like conditions for the existence of algebras of fractions of linear Jordan algebras, see [J. Algebra 237, 798–812 (2001; Zbl 1006.17024)]. Following theses ideas, several authors began to study rings or algebras of quotients in the nonassociative setting. Among others, M. Siles Molina introduced a notion of algebras of quotients in the Lie setting [J. Pure Appl. Algebra 188, 175–188 (2004; Zbl 1036.17006)], E. García and M. Gómez-Lozano defined a notion of Martindale-like quotients in the Jordan setting [J. Pure Appl. Algebra 194, 127–145 (2004; Zbl 1127.17027)], and J. Bowling and K. McCrimmon gave a quadratic extension of Martínez’ paper in [J. Algebra 312, 56–63 (2007; Zbl 1169.17022)]. In this paper, the authors relate the notion of Lie algebra of quotients with certain associative quotients:

(1) When the Lie algebra is the symmetrization of an associative algebra \(R\) (if \(R\) is associative, \(R^-\) with product \([x,y]:=xy-yx\) is a Lie algebra) and \(R\subset Q\subset Q_s(R)\) where \(Q_s(R)\) denotes the Martindale symmetric ring of quotients of \(R\), the authors prove that \(Q^-/Z(Q)\) is a Lie algebra of quotients of \(R^-/Z(R)\) (where \(Z(\;)\) denotes the center).

(2) If \(L\) is a Lie algebra with \(Z(L)=0\), \(L\) can be considered as a subalgebra of \(\text{End}(L)^-\) via the adjoint representation. If \(Q\) is a Lie algebra of quotients of \(L\), they show that the associative subalgebra \(A(Q)\) of \(\text{End}(Q)^-\) generated by \(Q\) is a left quotient algebra of \(A_0=\{\mu\in A(Q)\mid \mu(L)\subset L\}\) which contains all the elements \(\text{ad}_x\), \(x\in L\).

(1) When the Lie algebra is the symmetrization of an associative algebra \(R\) (if \(R\) is associative, \(R^-\) with product \([x,y]:=xy-yx\) is a Lie algebra) and \(R\subset Q\subset Q_s(R)\) where \(Q_s(R)\) denotes the Martindale symmetric ring of quotients of \(R\), the authors prove that \(Q^-/Z(Q)\) is a Lie algebra of quotients of \(R^-/Z(R)\) (where \(Z(\;)\) denotes the center).

(2) If \(L\) is a Lie algebra with \(Z(L)=0\), \(L\) can be considered as a subalgebra of \(\text{End}(L)^-\) via the adjoint representation. If \(Q\) is a Lie algebra of quotients of \(L\), they show that the associative subalgebra \(A(Q)\) of \(\text{End}(Q)^-\) generated by \(Q\) is a left quotient algebra of \(A_0=\{\mu\in A(Q)\mid \mu(L)\subset L\}\) which contains all the elements \(\text{ad}_x\), \(x\in L\).