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Let consider $L$ be a Leibniz algebra which is left nilpotent. ( I do not know what is left nilpotent of class 3).

A Leibniz algebra L is said to be nilpotent, if for lower central series there exists n ∈ N such that $L^{n} = 0$. The minimal number $n$ with this property is said to be index of nilpotency of algebra $L$. Is the class of nilpotency the same with index of nilpotency?

An n-dimensional Leibniz algebra is called null filiform if $dim L^{i}=n+1-i$, where $1 \leq i \leq n+1$. Is a left nilpotent Leibniz algebra of class 3 a null filiform Leibniz algebras?

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An algebra $A$ is called left Leibniz, if its multiplication satisfies the left Leibniz identity $$ (ab) c = a (bc) − b (ac) $$ for all $a, b, c ∈ A$. The $2$-sided ideals $A^i$ and $A_i$ are defined by $A^1=A$ and $A^{i+1}=AA^i$, and $A_1=A$, $A_{i+1}=\sum_{p=1}^iA_pA_{i+1-p}$, from the left. Then $A$ is called left nilpotent if $A_n=0$ for some $n$. One can show that $A_n=A^n$ for all $n\ge 1$.

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  • $\begingroup$ So the index of nilpotency is the same with the class of nilpotency? Since in some papers I saw these two words! $\endgroup$ – user40491 Apr 22 at 18:44
  • $\begingroup$ Yes, it is the same. $\endgroup$ – Dietrich Burde Apr 22 at 18:45
  • $\begingroup$ According to your detailed definition, we can say that a left nilpotent Leibniz of class 3 is null filiform, am I right? $\endgroup$ – user40491 Apr 22 at 18:48
  • $\begingroup$ What dimension has your Leibniz algebra of class $3$? $\endgroup$ – Dietrich Burde Apr 22 at 18:57
  • $\begingroup$ In fact, we have a variety of Leibniz algebras satisfying a special polynomial identity in such a way the Leibniz algebras satisfying the identity are left nilpotent of class 3. $\endgroup$ – user40491 Apr 22 at 19:24

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