# what is a left nilpotent Leibniz algebra

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?

## 1 Answer

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 $$n\ge 1$$.

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