Formal Lie groupoids
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-algebras are a higher generalization (a “vertical categorification”) of Lie algebras: in an -algebra the Jacobi identity is allowed to hold (only) up to higher coherent homotopy.
An -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 -algebra is a Lie ∞-algebroid with a single object.
-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 -algebra “exponentiates” to a smooth ∞-group .
Abstract definition in terms of algebras over an operad
An -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.
Original definition in terms of lots of brackets
Originally -algebras were defined as follows:
an -algebra is an -graded vector space together with for each a list of graded-symmetric linear maps
of degree -1, that satisfy a generalized Jacobi identity of the form
for all elements , where the inner sum runs over all -unshuffles and where is the Koszul-signature of the permutation : the sign picked up by “unshuffling” according to .
In this form this appears for instance as (CattaneoSchaetz, (1)). Notice that there are different conventions on the gradings possible, which lead to similar formulas with different sign factors.
Notably for this says that the binary bracket , the trinary bracket and the unary “bracket” are related by
(Exercise for the reader: put in the right signs, depending on the degree of the ).
If the right hand side were 0, this would be the Jacobi identity of an ordinary Lie algebra (or super Lie algebra, rather). So the image under of the trinary bracket in the -algebra is a measure for how the ordinary Jacobi identity fails in an -algebra.
But the point is that it does not just fail: it fails by the specific homotopy expressed by , and this homotopy itself is coherent, in that it satisfies suitable relatins itself, obtained by evaluating the above sum expression for .
Reformulation in terms of semifree differential coalgebra
A little later it was realized that the above huge sum expressions above just expresses the fact that the differential in a semifree dg-coalgebra squares to 0, :
An -algebra is
an -graded vector space ;
equipped with a differential of degree on the free graded co-commutative coalgebra over that squares to 0
Here the free graded co-commutative co-algebra is, as a vector space, the same as the graded Grassmann algebra whose elements we write as
etc (where the is just a funny way to write the 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 as
where each acts as when evaluated on a homogeneous element of the form and is then uniquely extended to all of by extending it as a coderivation on a coalgebra.
For instance 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 squares to 0 is precisely equivalent to the infinite tower of generalized Jacobi identities:
So in conclusion we have:
An -algebra is a dg-coalgebra whose underlying coalgebra is cofree and concentrated in negative degree.
Reformulation in terms of semifree differential algebra
The reformulation of an -algebra as simply a semi-co-free graded-co-commutative coalgebra 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 is degreewise finite dimensional, we can simply pass to its degreewise dual graded vector space . Its Grassmann algebra is then naturally equipped with an ordinary differential which acts on as
When the grading-dust has settled one finds that with
with the ground field in degree 0, the degree 1-elements of in degree 1, etc, that is of degree +1 and of course squares to 0
This means that we have a semifree dga
In the case that happens to be an ordinary Lie algebra, this is the ordinary Chevalley-Eilenberg algebra of this Lie algebra. Hence we should generally call the Chevalley-Eilenberg algebra of the -algebra .
One observes that this construction is bijective: every (degreewise finite dimensional) cochain semifree dga generated in positive degree comes from a (degreewise finite dimensional) -algebra this way.
This means that we may just as well define a (degreewise finite dimensional) -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 -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 -algebra structure on is equivalently a dg-algebra-structure on .
Let be a degreewise finite-dimensional graded vector space equipped with multilinear graded-symmetric maps
of degree -1, for each .
Let be a basis of and a dual basis of the degreewise dual . Equip the Grassmann algebra with a derivation
defined on generators by
Here we take to be of the same degree as . Therefore this derivation has degree +1.
We compute the square :
Here the wedge product on the right projects the nested bracket onto its graded-symmetric components. This is produced by summing over all permutations weighted by the Koszul-signature of the permutation:
The sum over all permutations decomposes into a sum over the -unshuffles and a sum over permutations that act inside the first and the last indices. By the graded-symmetry of the bracket, the latter do not change the value of the nested bracket. Since there are many of them, we get
Therefore the condition is equivalent to the condition
for all and all . This is equation (1) which says that is an -algebra.
In terms of algebras over an operad
-algebras are precisely the algebras over an operad of the cofibrant resolution of the Lie operad.
An -algebra for which is concentrated in the first degree is a Lie -algebra (sometimes also: ”-algebra”).
An -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 -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 -algebra generated just in degree 1 is an ordinary Lie algebra ;
an -algebra generated just in degree 1 and 2 is a Lie 2-algebra ;
an -algebra generated just in degree 1, 2 and 3 is a Lie 3-algebra ;
if is a Lie algebra over , and is the complex consisting of the field in degree , then an -algebra morphism from to is precisely a degree Lie algebra cocycle.
The skew-symmetry of the Lie bracket is retained strictly in -algebras. It is expected that weakening this, too, yields a more general vertical categorification of Lie algebras. For this has been worked out by Dmitry Roytenberg: On weak Lie 2-algebras.
The horizontal categorification of -algebras are -algebroids.
An -algebra with only non-vanishing is called an n-Lie algebra – to be distinguished from a Lie -algebra ! However, in large parts of the literature -Lie algebras are considered for which is not of the required homogeneous degree in the grading, or in which no grading is considered in the first place. Such -Lie algebras are not special examples of -algebras, then. For more see n-Lie algebra.
An -alghebra internal to super vector spaces is a super L-∞ algebra.
Classes of examples
Model category structure
See model structure for L-∞ algebras.
Relation to dg-Lie algebras
Every dg-Lie algebra is in an evident way an -algebra. Dg-Lie algebras are precisely those -algebras for which all -ary brackets for are trivial. These may be thought of as the strict -algebras: those for which the Jacobi identity holds on the nose and all its possible higher coherences are trivial.
Let be a field of characteristic 0 and write for the category of -algebras over .
Then every object of is quasi-isomorphic to a dg-Lie algebra.
Moreover, one can find a functorial replacement: there is a functor
such that for each
is a dg-Lie algebra;
there is a quasi-isomorphism
This appears for instance as (KrizMay, cor. 1.6).
Relation to -Lie groupoids
In generalization to how a Lie algebra integrates to a Lie group, -algebras integrate to ∞-Lie groups.
Lie integrated ∞-Lie groupoids.
The original references are:
A quick web entry is:
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:
- John Baez and Alissa Crans, Higher-dimensional algebra VI: Lie 2-algebras, TAC 12, (2004), 492–528. (arXiv)
For more general ‘weak Lie 2-algebras’, see:
Equipped with bilinear invariant polynomial
- Yunfeng Jiang, Motivic Milnor fiber of cyclic -algebras (arXiv:0909.2858)