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I'm trying to formally understand the free Gerstenhaber algebra on a set $X = \{x_1,...,x_n\}$. I've just started working with graded objects and I am not used to the complexity of structures such as Gerstenhaber algebras. I don't really have any references for this, so I thought I'd ask here.

I am familiar with other, simpler free constructions, such as the free associative algebra on $X$. This can be constructed by either taking the free vector space $V(X)$ with basis $X$ and then taking the tensor algebra $T(V(X))$, or alternatively by taking the free monoid $M(X)$ on $X$ and then taking the free vector space $V(M(X))$ with basis $M(X)$.

However, if I try to do something similar with the Gernstenhaber algebra, since I need to have two operations, I would need to consider words that are a combination of multiplication and bracket, so they would look for example like $[[x_1,x_3],x_4x_2]x_1$. Then I might define two products on these words, one being concatenation and the other being bracketing, then take the free vector space on this and take quotients to get the relations needed on the multiplication and bracket. Somehow degree will need to come in to play, but I don't understand how. I'm not sure if I'm close or not, but somehow a clear understanding of what to do is evading me. Can someone explain how to do this formally or provide a good reference?

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Are you familiar with the free Lie algebra? – Qiaochu Yuan Aug 8 '12 at 3:56
up vote 0 down vote accepted

Ok I think I figured out how to do it. First I realize we need to specify degrees of the elements of $X$. I want them to all be degree $0$. Then I guess by free object we mean an object such that all degree preserving set maps from $X$ to a Gerstenhaber algebra extend to Gerstenhaber algebra maps. (It is sort of bothering me that we have this restriction to degree preserving maps but it seems necessary).
Start with the free monoid $M(X)$. Call $A_0 = M(X)$. Then for $n \leq -1$ let $A_n$ be the set consisting of all possible $|n|$ formal brackets $[a,b]$ where $a,b \in M(X)$. So $A_5$ would be elements like $[[a,b],[c,[d,e]]]$ where $a,b,c,d,e \in M(X)$. Call $A_n$ the elements of degree $n$. I'll call graded sets with degree $0$ associative multiplication and degree $-1$ bracket "Gerstenhaber monoids" and define morphisms to be degree preserving maps that also preserve multiplication and bracket. Then $A(X) = (A_n)$ should be a free object in this category on $X$ with all elements degree $0$ . Define $V(A(X))$ to be the vector space with basis $A(X)$. Define multiplication and bracket via distributive law so that they are bi-linear to give this sort of "pre-Gerstenhaber" algebra structure. Then take the quotient by the smallest ideal containing all relations. This should be the free Gerstenhaber algebra on $X$ where all elements of $X$ are degree $0$. I didn't check all the details but I think this should work.

If you have some insight into this I'm still interested.

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On the issue about degree-preserving maps, if you're in a graded category (of rings, modules, algebras, Lie algebras, Gerstenhaber algebras, or whatever) the morphisms are required to respect the grading. So that shouldn't worry you-it's just like only requiring group homomorphisms to factor through the free group. – Kevin Carlson Aug 8 '12 at 3:13

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