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Let $F[x,y]$ be a polynomial algebra space over field $F$, let define a Leibniz algebra structure on that. Is this defined Leibniz algebra on $F[x,y]$ is free Leibniz algebra? How can we check this is free?

Nil
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1 Answers1

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There is a Leibniz algebra structure on $\mathbb{F}[x,y]$ as follows:

$$[f,g]=h, \textrm{ where } h(x,y)=f(x,x)(g(y,y)-g(x,x))$$

but I doubt that it is free.

Reference: Lei L., Yun Zh. $F[x,y]$ as a Dialgebra and a Leibniz Algebra. Communications in Algebra. Vol. 38(9), 2010, 3417-3447.

Rustam Turdibaev
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