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A Lie algebra over $\mathbb Z$ is defined to be an abelian group with a bracketing operation $[\cdot,\cdot]$, satisfying the Jacobi identity and the relation $[x,x]=0$ for every $x$. On the other hand, a quasi-Lie algebra is defined by replacing the axiom $[x,x]=0$ with the antisymmetry axiom $[x,y]+[y,x]=0$ for all $x$ and $y$. Jerry Levine used the free quasi-Lie algebra to study the group of homology cylinders. See this paper and this paper. My question is whether there are other contexts in which quasi-Lie algebras appear. Lie algebras arise naturally in all sorts of different contexts, but I don't know of many places in which quasi-Lie algebras appear. For example, if one looks at the associated graded object for the lower central series of a group, one naturally obtains a Lie algebra and not a quasi-Lie algebra.

One possible answer is this MO post, where it is noted that in operad language, the quasi-Lie axioms are necessary, and this is quite germane to why Levine needed to use quasi-Lie algebras, since he was essentially building a group out of the Lie operad. But I'd like to know of any other examples where quasi-Lie algebras occur that people might know.

Added (Moskovich): I wonder also whether Levine was the first person ever to consider quasi-Lie algebras, in 2002. Surely that can't possibly be the case- as a structure, it's surely too natural to be so new!

Added (Conant) This is not just an idle question. My main motivation for asking this question is to find new link concordance invariants. Milnor's mu invariants are associated to the lower central series of a link group, the associated graded object being a Lie algebra. I'm hoping there is some kind of quasi-Lie algebra one can assign to a link complement that generalizes this associated graded Lie algebra.

Edit: Based on the comments and Mariano's answer, I'd like to make some clarifications. First, when working over the integers, maybe it is better to use the term "Lie ring" as opposed to "Lie algebra," but I had just been following Levine's convention. (Unfortunately, Hilton's definition of a quasi-Lie ring, given in the reference in Mariano's answer, does not match Levine's!) Levine's definition means "a graded (or super) Lie algebra where all elements are of even degree." So one could take the even degree part of the Lie algebra formed from homotopy groups under the Whitehead product, as Tom mentions.

Edit: It turns out that Levine's definition of a quasi-Lie algebra is a specific case of Hilton's definition (see Mariano's answer), where the module $M$ is $0$.

6 added 624 characters in body

A Lie algebra over $\mathbb Z$ is defined to be an abelian group with a bracketing operation $[\cdot,\cdot]$, satisfying the Jacobi identity and the relation $[x,x]=0$ for every $x$. On the other hand, a quasi-Lie algebra is defined by replacing the axiom $[x,x]=0$ with the antisymmetry axiom $[x,y]+[y,x]=0$ for all $x$ and $y$. Jerry Levine used the free quasi-Lie algebra to study the group of homology cylinders. See this paper and this paper. My question is whether there are other contexts in which quasi-Lie algebras appear. Lie algebras arise naturally in all sorts of different contexts, but I don't know of many places in which quasi-Lie algebras appear. For example, if one looks at the associated graded object for the lower central series of a group, one naturally obtains a Lie algebra and not a quasi-Lie algebra.

One possible answer is this MO post, where it is noted that in operad language, the quasi-Lie axioms are necessary, and this is quite germane to why Levine needed to use quasi-Lie algebras, since he was essentially building a group out of the Lie operad. But I'd like to know of any other examples where quasi-Lie algebras occur that people might know.

Added (Moskovich): I wonder also whether Levine was the first person ever to consider quasi-Lie algebras, in 2002. Surely that can't possibly be the case- as a structure, it's surely too natural to be so new!

Added (Conant) This is not just an idle question. My main motivation for asking this question is to find new link concordance invariants. Milnor's mu invariants are associated to the lower central series of a link group, the associated graded object being a Lie algebra. I'm hoping there is some kind of quasi-Lie algebra one can assign to a link complement that generalizes this associated graded Lie algebra.

Edit: Based on the comments and Mariano's answer, I'd like to make some clarifications. First, when working over the integers, maybe it is better to use the term "Lie ring" as opposed to "Lie algebra," but I had just been following Levine's convention. (Unfortunately, Hilton's definition of a quasi-Lie ring, given in the reference in Mariano's answer, does not match Levine's!) Levine's definition means "a graded (or super) Lie algebra where all elements are of even degree." So one could take the even degree part of the Lie algebra formed from homotopy groups under the Whitehead product, as Tom mentions.

5 added 416 characters in body

A Lie algebra over $\mathbb Z$ is defined to be an abelian group with a bracketing operation $[\cdot,\cdot]$, satisfying the Jacobi identity and the relation $[x,x]=0$ for every $x$. On the other hand, a quasi-Lie algebra is defined by replacing the axiom $[x,x]=0$ with the antisymmetry axiom $[x,y]+[y,x]=0$ for all $x$ and $y$. Jerry Levine used the free quasi-Lie algebra to study the group of homology cylinders. See this paper and this paper. My question is whether there are other contexts in which quasi-Lie algebras appear. Lie algebras arise naturally in all sorts of different contexts, but I don't know of many places in which quasi-Lie algebras appear. For example, if one looks at the associated graded object for the lower central series of a group, one naturally obtains a Lie algebra and not a quasi-Lie algebra.

One possible answer is this MO post, where it is noted that in operad language, the quasi-Lie axioms are necessary, and this is quite germane to why Levine needed to use quasi-Lie algebras, since he was essentially building a group out of the Lie operad. But I'd like to know of any other examples where quasi-Lie algebras occur that people might know.

Added (Moskovich): I wonder also whether Levine was the first person ever to consider quasi-Lie algebras, in 2002. Surely that can't possibly be the case- as a structure, it's surely too natural to be so new!

Added (Conant) This is not just an idle question. My main motivation for asking this question is to find new link concordance invariants. Milnor's mu invariants are associated to the lower central series of a link group, the associated graded object being a Lie algebra. I'm hoping there is some kind of quasi-Lie algebra one can assign to a link complement that generalizes this associated graded Lie algebra.