How to construct a Free Leib algebras with example

Asked Jun 13 '17 at 17:12

Active Jun 13 '17 at 17:12

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According to the Examples of Free Lie Algebra we have a useful example for free Lie algebra. Now I want to construct an example for free Leibniz algebra. I faced an example for free Leibniz algebra as follows, but I need your guidance to build free Leibniz algebra on $X=\{x_{1} , x_{2}\}$.

Example: Let we have $\{a,b \}$ such that $[b,a]=a$ and $[b,b]=a$, then it is satisfying Leibniz identity such that it is a left Leibniz algebra.

asked Jun 13 '17 at 17:12

Nil

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Why not taking the free algebra generated by $x_1,x_2$ quotiented by the Jacobi relation ? – Jun 13 '17 at 17:18
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The construction of a free Leibniz algebra is given here. – Dietrich Burde Jun 13 '17 at 19:08
@N.H. you're right that proving the existence is trivial, but maybe the question is rather to compute a basis. The word "construct" is somewhat vague here. – YCor Jun 13 '17 at 23:19
@YCor : This make indeed more sense as a question. – Jun 14 '17 at 06:56

How to construct a Free Leib algebras with example

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