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Un théorème de Poincaré-Birkhoff-Witt pour les algèbres de Leibniz. (A Poincaré-Birkhoff-Witt theorem for Leibniz algebras). (French) Zbl 1020.17002

Leibniz algebras are characterized by the property that the multiplication (called a bracket) from the right is a derivation but the bracket is not necessarily skew-symmetric as for Lie algebras. Dialgebras have two binary multiplications \(\dashv\) and \(\vdash\) subject to some associativity conditions. They produce Leibniz algebras in the same way as one obtains Lie algebras from associative algebras: If \((D,\vdash,\dashv)\) is a dialgebra, then \((D,[x,y]=x\dashv y-y\vdash x)\) is a Leibniz algebra. If \(K\) is a commutative ring with 1 and \(V\) is a free \(K\)-module, then, as a \(K\)-module, the free dialgebra \(DL(V)\) is spanned by the elements \(x_1\vdash\cdots\vdash x_p\vdash y\dashv z_1\dashv\cdots\dashv z_q\), \(x_i,y,z_j\in V\), and is isomorphic to the tensor product \(T(V)\otimes V\otimes T(V)\), where \(T(V)\) is the tensor algebra of \(V\). If \(\mathbf{g}\) is a Leibniz algebra, then the factor algebra \(Ud(\mathbf{g})\) of \(DL(\mathbf{g})\) modulo the ideal generated by all \([x,y]-(x\dashv y-y\vdash x)\), \(x,y\in\mathbf{g}\), is the universal enveloping dialgebra of \(\mathbf{g}\).
In the paper under review the authors prove a Leibniz analogue of the Poincaré-Birkhoff-Witt theorem: If \(\mathbf{g}\) is a Leibniz algebra which is a free \(K\)-module and \(\mathbf{g}_{Lie}\) is the associated Lie algebra, then the associated graded dialgebra \(grUd(\mathbf{g})\) of \(Ud(\mathbf{g})\) is isomorphic to the tensor product \(S(\mathbf{g}_{Lie})\otimes\text\textbf{g}\), where \(S(\mathbf{g}_{Lie})\) is the symmetric algebra of \(\mathbf{g}_{Lie}\). The proof of the authors is inspired by the classical proof of the Poincaré-Birkhoff-Witt theorem for Lie algebras. As the authors mention, although not explicitly stated, the Leibniz analogue of the Poincaré-Birkhoff-Witt theorem can be also derived from the results in [J. L. Loday and T. Pirashvili, Georgian Math. J. 5, 263-276 (1998; Zbl 0909.18003)].

MSC:

17A32 Leibniz algebras
17B35 Universal enveloping (super)algebras
17A30 Nonassociative algebras satisfying other identities

Citations:

Zbl 0909.18003
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References:

[1] Bourbaki N., Groupes et algébres de Lie (1971)
[2] Bourbaki N., Algèbre (1970) · Zbl 0455.18010
[3] Cuvier C., Ann. Sc. Ecol. Norm. Sup. 4me série, t. 27 pp 1– (1994)
[4] Cohn P. M., Universal Algebra 6 (1981)
[5] Loday J. L., Grund. Math. Wiss. 301 (1992)
[6] Loday J. L., L’Enseignement mathématique 39 pp 269– (1993)
[7] Loday J. L., I, in: Algèbres ayant deux opérations associatives (digèbres) pp 141– (1995)
[8] DOI: 10.1007/BF01445099 · Zbl 0821.17022 · doi:10.1007/BF01445099
[9] Loday J. L., Georgian Math. J. 5 pp 263– (1998) · Zbl 0909.18003 · doi:10.1023/B:GEOR.0000008125.26487.f3
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