symmetric monoidal (∞,1)-category of spectra
and
$L_\infty$-algebras (or strong homotopy Lie algebras) are a higher generalization (a “vertical categorification”) of Lie algebras: in an $L_\infty$-algebra the Jacobi identity is allowed to hold (only) up to higher coherent homotopy.
An $L_\infty$-algebra that is concentrated in lowest degree is an ordinary Lie algebra. If it is concentrated in the lowest two degrees is is a Lie 2-algebra, etc.
From another perspective: an $L_\infty$-algebra is a Lie ∞-algebroid with a single object.
$L_\infty$-algebras are infinitesimal approximations of smooth ∞-groups in analogy to how an ordinary Lie algebra is an infinitesimal approximation of a Lie group. Under Lie integration every $L_\infty$-algebra $\mathfrak{g}$ “exponentiates” to a smooth ∞-group $\Omega \exp(\mathfrak{g})$.
An $L_\infty$-algebra is an algebra over an operad in the category of chain complexes over the L-∞ operad.
In the following we spell out in detail what this means in components.
For $V$ a graded vector space, for $v_i \in V_{\vert v_i\vert}$ homogenously graded elements, and for $\sigma$ a permutation of $n$ elements, write $\chi(\sigma, v_1, \cdots, v_n)\in \{-1,+1\}$ for the product of the signature of the permutation with a factor of $(-1)^{\vert v_i \vert \vert v_j \vert}$ for each interchange of neighbours $(\cdots v_i,v_j, \cdots )$ to $(\cdots v_j,v_i, \cdots )$ involved in the permutation.
An $L_\infty$-algebra is
a graded vector space $V$;
for each $n \in \mathbb{N}$ a multilinear map called the $n$-ary bracket
$l_n(\cdots) \coloneqq [-,-, \cdots, -] \colon V^{\wedge n} \to V$
of degree $n-2$
such that
each $l_n$ is graded antisymmetric, in that for every permutation $\sigma$ and homogeneously graded elements $v_i \in V_{\vert v_i \vert}$ then
the generalized Jacobi identity holds:
for all $n$, all and homogeneously graded elements $v_i \in V_i$ (here the inner sum runs over all $(i,j)$-unshuffles $\sigma$).
There are various different conventions on the gradings possible, which lead to similar formulas with different signs.
In lowest degrees the generalized Jacobi identity says that
for $n = 1$: the unary map $\partial \coloneqq l_1$ squares to 0:
1: for $n = 2$: the unary map $\partial$ is a graded derivation of the binary map
hence
When all higher brackets vanish, $l_{k \gt 2}= 0$ then for $n = 3$:
this is the graded Jacobi identity. So in this case the $L_\infty$-algebra is equivalently a dg-Lie algebra.
When $l_3$ is possibly non-vanishing, then on elements $x_i$ on which $\partial = l_1$ vanishes then the generalized Jacobi identity for $n = 3$ gives
This shows that the Jacobi identity holds up to an “exact” term, hence up to homotopy.
A little later it was realized that the above huge sum expressions above just expresses the fact that the differential $D$ in a semifree dg-coalgebra squares to 0, $D^2 = 0$:
An $L_\infty$-algebra is
an $\mathbb{N}_+$-graded vector space $\mathfrak{g}$;
equipped with a differential $D : \vee^\bullet \mathfrak{g} \to \vee^\bullet \mathfrak{g}$ of degree $-1$ on the free graded co-commutative coalgebra over $\mathfrak{g}$ that squares to 0
Here the free graded co-commutative co-algebra $\vee^\bullet \mathfrak{g}$ is, as a vector space, the same as the graded Grassmann algebra $\wedge^\bullet \mathfrak{g}$ whose elements we write as
etc (where the $\vee$ is just a funny way to write the wedge $\wedge$, in order to remind us that:…)
but throught of as equipped with the standard coproduct
(work out or see the references for the signs and prefacors).
Since this is a free graded co-commutative coalgebra, one can see that any differential
on it is fixed by its value “on cogenerators” (a statement that is maybe unfamiliar, but simply the straightforward dual of the more familar statement to which we come below, that differentials on free graded algebras are fixed by their action on generators) which means that we can decompose $D$ as
where each $D_i$ acts as $l_i$ when evaluated on a homogeneous element of the form $t_1 \vee \cdots \vee t_n$ and is then uniquely extended to all of $\vee^\bullet \mathfrak{g}$ by extending it as a coderivation on a coalgebra.
For instance $D_2$ acts on homogeneous elements of word lenght 3 as
exercise for the reader: spell this all out more in detail with all the signs and everyrthing. Possibly by looking it up in the references given below.
Using this, one checks that the simple condition that $D$ squares to 0 is precisely equivalent to the infinite tower of generalized Jacobi identities:
So in conclusion we have:
An $L_\infty$-algebra is a dg-coalgebra whose underlying coalgebra is cofree and concentrated in negative degree.
The reformulation of an $L_\infty$-algebra as simply a semi-co-free graded-co-commutative coalgebra $(\vee^\bullet \mathfrak{g}, D)$ is a useful repackaging of the original definition, but the coalgebraic aspect tends to be not only unfamiliar, but also a bit inconvenient. At least when the graded vector space $\mathfrak{g}$ is degreewise finite dimensional, we can simply pass to its degreewise dual graded vector space $\mathfrak{g}^*$. Its Grassmann algebra $\wedge^\bullet \mathfrak{g}^*$ is then naturally equipped with an ordinary differential $d = D^*$ which acts on $\omega \in \wedge^\bullet \mathfrak{g}^*$ as
When the grading-dust has settled one finds that with
with the ground field in degree 0, the degree 1-elements of $\mathfrak{g}^*$ in degree 1, etc, that $d$ is of degree +1 and of course squares to 0
This means that we have a semifree dga
In the case that $\mathfrak{g}$ happens to be an ordinary Lie algebra, this is the ordinary Chevalley-Eilenberg algebra of this Lie algebra. Hence we should generally call $CE(\mathfrak{g})$ the Chevalley-Eilenberg algebra of the $L_\infty$-algebra $\mathfrak{g}$.
One observes that this construction is bijective: every (degreewise finite dimensional) cochain semifree dga generated in positive degree comes from a (degreewise finite dimensional) $L_\infty$-algebra this way.
This means that we may just as well define a (degreewise finite dimensional) $L_\infty$-algebra as an object in the opposite category of (degreewise finite dimensional) commutative dg-algebras that are semifree dgas and generated in positive degree.
And this turns out to be one of the most useful perspectives on $L_\infty$-algebras.
In particular, if we simply drop the condition that the dg-algebra be generated in positive degree and allow it to be generated in non-negative degree over the algebra in degree 0, then we have the notion of the (Chevalley-Eilenberg algebra of) an L-infinity-algebroid.
We discuss in explit detail the computation that shows that an $L_\infty$-algebra structure on $\mathfrak{g}$ is equivalently a dg-algebra-structure on $\wedge^\bullet \mathfrak{g}^*$.
Let $\mathfrak{g}$ be a degreewise finite-dimensional $\mathbb{N}_+$graded vector space equipped with multilinear graded-symmetric maps
of degree -1, for each $k \in \mathbb{N}_+$.
Let $\{t_a\}$ be a basis of $\mathfrak{g}$ and $\{t^a\}$ a dual basis of the degreewise dual $\mathfrak{g}^*$. Equip the Grassmann algebra $Sym^\bullet \mathfrak{g}^*$ with a derivation
defined on generators by
Here we take $t^a$ to be of the same degree as $t_a$. Therefore this derivation has degree +1.
We compute the square $d^2 = d \circ d$:
Here the wedge product on the right projects the nested bracket onto its graded-symmetric components. This is produced by summing over all permutations $\sigma \in \Sigma_{k+l-1}$ weighted by the Koszul-signature of the permutation:
The sum over all permutations decomposes into a sum over the $(l,k-1)$-unshuffles and a sum over permutations that act inside the first $l$ and the last $(k-1)$ indices. By the graded-symmetry of the bracket, the latter do not change the value of the nested bracket. Since there are $(k-1)! l!$ many of them, we get
Therefore the condition $d^2 = 0$ is equivalent to the condition
for all $n \in \mathbb{N}$ and all $\{t_{a_i} \in \mathfrak{g}\}$. This is equation (1) which says that $\{\mathfrak{g}, \{[-,\dots,-]_k\}\}$ is an $L_\infty$-algebra.
$L_\infty$-algebras are precisely the algebras over an operad of the cofibrant resolution of the Lie operad.
An $L_\infty$-algebra for which $V$ is concentrated in the first $n$ degree is a Lie $n$-algebra (sometimes also: “$L_n$-algebra”).
An $L_\infty$-algebra for which only the unary operation and the binary bracket are non-trivial is a dg-Lie algebra: a Lie algebra internal to the category of dg-algebras. From the point of view of higher Lie theory this is a strict $L_\infty$-algebra: one for which the Jacobi identity does happen to hold “on the nose”, not just up to nontrivial coherent isomorphisms.
So in particular
an $L_\infty$-algebra generated just in degree 1 is an ordinary Lie algebra ;
an $L_\infty$-algebra generated just in degree 1 and 2 is a Lie 2-algebra ;
an $L_\infty$-algebra generated just in degree 1, 2 and 3 is a Lie 3-algebra ;
if $\mathfrak{g}$ is a Lie algebra over $\mathbf{K}$, and $b^{k-1}\mathbb{K}$ is the complex consisting of the field $\mathbb{K}$ in degree $1-k$, then an $L_\infty$-algebra morphism from $\mathfrak{g}$ to $b^{k-1}\mathbb{K}$ is precisely a degree $k$ Lie algebra cocycle.
The skew-symmetry of the Lie bracket is retained strictly in $L_\infty$-algebras. It is expected that weakening this, too, yields a more general vertical categorification of Lie algebras. For $n=2$ this has been worked out by Dmitry Roytenberg: On weak Lie 2-algebras.
The horizontal categorification of $L_\infty$-algebras are $L_\infty$-algebroids.
An $L_\infty$-algebra with only $D_n$ non-vanishing is called an n-Lie algebra – to be distinguished from a Lie $n$-algebra ! However, in large parts of the literature $n$-Lie algebras are considered for which $D_n$ is not of the required homogeneous degree in the grading, or in which no grading is considered in the first place. Such $n$-Lie algebras are not special examples of $L_\infty$-algebras, then. For more see n-Lie algebra.
An $L_\infty$-alghebra internal to super vector spaces is a super L-∞ algebra.
automorphism Lie 2-algebra?
For every $\infty$-Lie algebra $\mathfrak{g}$ there is its automorphism ∞-Lie algebra. In terms of rational homotopy theory this models the rational automorphism group of the rational space corresponding to $\mathfrak{g}$.
Heisenberg Lie n-algebra of an n-plectic manifold or more generally of an n-plectic smooth infinity-groupoid
See model structure for L-∞ algebras.
Every dg-Lie algebra is in an evident way an $L_\infty$-algebra. Dg-Lie algebras are precisely those $L_\infty$-algebras for which all $n$-ary brackets for $n \gt 2$ are trivial. These may be thought of as the strict $L_\infty$-algebras: those for which the Jacobi identity holds on the nose and all its possible higher coherences are trivial.
Let $k$ be a field of characteristic 0 and write $L_\infty Alg_k$ for the category of $L\infty$-algebras over $k$.
Then every object of $L_\infty Alg_k$ is quasi-isomorphic to a dg-Lie algebra.
Moreover, one can find a functorial replacement: there is a functor
such that for each $\mathfrak{g} \in L_\infty Alg_k$
$W(\mathfrak{k})$ is a dg-Lie algebra;
there is a quasi-isomorphism
This appears for instance as (KrizMay, cor. 1.6).
In generalization to how a Lie algebra integrates to a Lie group, $L_\infty$-algebras integrate to ∞-Lie groups.
See
and
Lie integrated ∞-Lie groupoids.
$L_\infty$-algebra
The original references are:
Tom Lada, Jim Stasheff, Introduction to sh Lie algebras for physicists, Int. J. Theo. Phys. 32 (1993), 1087–1103. (arXiv:hep-th/9209099)
Tom Lada, Martin Markl, Strongly homotopy Lie algebras Communications in Algebra Volume 23, Issue 6, (1995) (arXiv:hep-th/9406095)
A quick web entry is:
See also
A discussion in terms of resolutions of the Lie operad is for instance in
See also for instance section 3.1 of:
A detailed reference for Lie 2-algebras is:
For more general ‘weak Lie 2-algebras’, see: