On the commutator in Leibniz algebras

A. Dzhumadil'daev, N. Ismailov, B. Sartayev
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引用次数: 2

Abstract

We prove that the class of algebras embeddable into Leibniz algebras with respect to the commutator product is not a variety. It is shown that every commutative metabelain algebra is embeddable into Leibniz algebras with respect to the anti-commutator. Furthermore, we study polynomial identities satisfied by the commutator in every Leibniz algebra. We extend the result of Dzhumadil’daev in [A. S. Dzhumadil’daev, [Formula: see text]-Leibniz algebras, Serdica Math. J. 34(2) (2008) 415–440]. to identities up to degree 7 and give a conjecture on identities of higher degrees. As a consequence, we obtain an example of a non-Spechtian variety of anticommutative algebras.
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莱布尼兹代数中的对易子
证明了对易子积可嵌入莱布尼茨代数的代数类不是变种。证明了对于反对子,每一个可交换元代数都可嵌入到莱布尼茨代数中。进一步,我们研究了在所有莱布尼兹代数中换位子满足的多项式恒等式。我们在[A]中推广了Dzhumadil 'daev的结果。S. Dzhumadil 'daev,[公式:见原文]-莱布尼茨代数,Serdica数学。J. 34(2)(2008) 415-440]。讨论7次以下的恒等式,并给出更高次恒等式的一个猜想。因此,我们得到了一个非spectex反交换代数的例子。
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