As \(L_{\infty}\)-algebras play a large role in my research, and more generally in mathematical physics, homotopy theory, modern geometry etc I thought it maybe useful to say a few words about them.

One should think of \(L_{\infty}\)-algebras as “homotopy relatives” of Lie algebras. In a sense I think of them as differential graded Lie algebras + “more”. I hope to make this a little clearer.

Definition: A supervector space \(V = V_{0} \oplus V_{1}\) is said to be an \(L_{\infty}\)-algebra if it comes equipped with a series of parity odd \(n\)-linear operations (\(n \geq 0\) ), which we denote as “brackets” \((, \cdots , )\) that

1) are symmetric \(( \bullet , \cdots, a, b , \cdots, \bullet) = (-1)^{\widetilde{a}\widetilde{b} }( \bullet , \cdots, b, a , \cdots, \bullet) \), \(a,b \in V\).

2) satisfy the homotopy Jacobi identities

\(\sum_{k+l=n-1} \sum_{(k,l)-\textnormal{unshuffels}}(-1)^{\epsilon}\left( (a_{\sigma(1)}, \cdots , a_{\sigma(k)}), a_{\sigma(k+1)}, \cdots, a_{\sigma(k+l)} \right)=0\)

hold for all \(n \geq 1\). Here \((-1)^{\epsilon}\) is a sign that arises due to the exchange of homogenous elements \(a_{i} \in V\). Recall that a \((k,l)\)-unshuffle is a permutation of the indices \(1, 2, \cdots k+l \) such that \(\sigma(1)\) < \(\cdots\) < \(\sigma(k)\) and \(\sigma(k+1)\) < \(\cdots \) < \(\sigma(k+l)\). The LHS of the above are referred to as Jacobiators.

So, we have a vector space with a series of brackets; \((\emptyset)\), \((a,b)\) , \((a,b,c)\) etc. If the zero bracket \((\emptyset)\) is zero then the \(L_{\infty}\)-algebra is said to be strict. Often the definition of \(L_{\infty}\)-algebra assumes this. With a non-vanishing zero bracket the algebra is often called “weak”, “with background” or “curved”.

Let us examine the first few Jacobi identities in order to make all this a little clearer. First let us assume a strict algebra and we will denote the one bracket as \(d\) (this will become clear).

1) \(d^{2}a = 0 \).

That is we have a differential graded algebra.

2) \(d (a,b) + (da, b) + (-1)^{\widetilde{a} \widetilde{b}} (db, a) =0\).

So the one bracket (the differential) satisfied a derivation rule over the 2-bracket.

3) \(d (a,b,c) + (da,b,c) + (-1)^{\widetilde{a} \widetilde{b}}(db, a, c) + (-1)^{\widetilde{c}(\widetilde{a} + \widetilde{b})} (dc, a, b)\)
\( + ((a,b), c) + (-1)^{\widetilde{b}\widetilde{c}}((a,c), b) + (-1)^{\widetilde{a}(\widetilde{b}+ \widetilde{c})} ((b,c), a)= 0\).

So we have the standard Jacobi identity up to something exact.

The higher Jacobi identities are not so easy to interpret in terms of things we all know. There are higher homotopy relations and thus the word “strong”. This should make it clearer what I mean by “differential graded Lie algebra + more”.

Note that the conventions here are not quite the same as originally used by Stasheff. In fact he used a \(\mathbb{Z}\)-grading where we use a \(\mathbb{Z}_{2}\)-grading. The brackets of Stasheff are skew-symmetric and (with superisation) they are even/odd parity for even/odd number of arguments. By employing the parity reversion function and including a few extra sign factors one can construct a series of brackets on \(\Pi V\) that are closer to Stasheff’s conventions, of course “superised”. This series of brackets on \(\Pi V\) then directly includes Lie superalgebras.

There are other “similarities” between Lie algebras and \(L_{\infty}\)-algebras. I may post more about some of these another time.

A few words about applications. \(L_{\infty}\)-algebras can be found behind the BV (BFV) formalism, deformation quantistion of Poisson manifolds and closed string field theory, for example.