Homotopy Loday algebras?

Let us work in the [math]\mathbb{Z}_{2}[/math] graded category.

A Loday algebra is a "crippled" or "wonky" Lie algebra. That is it satisfies the bracket satisfies the Jacobi identity, but is not skew symmetric.

An interesting question is if there exists a homotopy version.

I suspect that one can consider "crippled" [math]L_{\infty}[/math] -algebras. So, a series of brackets that satisfy higher Jacobi identities, but do not have a symmetry property.

Any body seen anything like that?

A differential over such an algebra would also be straight forward to define. I myself have defined notions of a differential [math]L_{\infty}[/math]-algebra and a differential BV-algebra. (We can talk more about these later if anyone is interested).

Cheers