symmetric monoidal (∞,1)-category of spectra
-algebras (or strong homotopy Lie 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.
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 .
In the following we spell out in detail what this means in components.
For a graded vector space, for homogenously graded elements, and for a permutation of elements, write for the product of the signature of the permutation with a factor of for each interchange of neighbours to involved in the permutation.
An -algebra is
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 : the unary map squares to 0:
1: for : the unary map is a graded derivation of the binary map
When all higher brackets vanish, then for :
When is possibly non-vanishing, then on elements on which vanishes then the generalized Jacobi identity for 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 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:
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 .
(Fully generally the following works when using not just dg-algebras but pro-objects in dg-algebras, see at model structure for L-infinity algebras – Use of pro-dg-algebras).
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 .
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.
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.
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.
automorphism Lie 2-algebra?
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.
Moreover, one can find a functorial replacement: there is a functor
such that for each
This appears for instance as (KrizMay, cor. 1.6).
The original references are:
A historical survey is
Marilyn Daily, -structures, PhD thesis, 2004 (web)
A detailed reference for Lie 2-algebras is:
For more general ‘weak Lie 2-algebras’, see:
Discussion of bilinear invariant polynomials on -algebras includes
The following lists, mainly in chronological order of their discovery, L-∞ algebra structures appearing in physics, notably in supergravity, BV-BRST formalism, string theory, higher Chern-Simons theory/AKSZ sigma-models and local field theory.
For more see also at higher category theory and physics.
In their equivalent formal dual guise of Chevalley-Eilenberg algebras (see above), -algebras of finite type – in fact super L-∞ algebras – appear in pivotal role in the D'Auria-Fré formulation of supergravity at least since
In the supergravity literature these CE-algebras are referred to as “FDA”s. This is short for “free differential algebra”, which is a slight misnomer for what in mathematics are called semifree dgas (or sometimes “quasi-free” dga-s).
Hisham Sati, Urs Schreiber, Jim Stasheff, example 5 in section 6.5.1, p. 54 of L-infinity algebra connections and applications to String- and Chern-Simons n-transport, in Quantum Field Theory, Birkhäuser (2009) 303-424 (arXiv:0801.3480)
Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields, International Journal of Geometric Methods in Modern Physics Volume 12, Issue 02 (2015) 1550018 (arXiv:1308.5264)
Further exposition of this includes
The understanding that the BV-BRST complex used in physics to model the derived critical locus of the action functionals of gauge theories is mathematically the formal dual Chevalley-Eilenberg algebra of a derived L-∞ algebroid originates around
Discussion in terms of homotopy Lie-Rinehart pairs is due to
The first explicit appearance of -algebras in theoretical physics is the -algebra structure on the BRST complex of the closed bosonic string found in the context of closed bosonic string field theory in
Jim Stasheff, Closed string field theory, strong homotopy Lie algebras and the operad actions of moduli space Talk given at the Conference on Topics in Geometry and Physics (1992) (arXiv:hep-th/9304061)
Next it was again -algebras of finite type that drew attention. It was eventually understood that the string structures which embody a refinement of the Green-Schwarz anomaly cancellation mechanism in heterotic string theory have a further smooth refinement as G-structures for the string 2-group, which is the Lie integration of a Lie 2-algebra called the string Lie 2-algebra. This is due to
and the relation to the Green-Schwarz mechanism is made explicit in
Ordinary Chern-Simons theory for a simple gauge group is all controled by a Lie algebra 3-cocycle. The generalization of Chern-Simons theory to AKSZ-sigma models was understood to be encoded by symplectic Lie n-algebroids (later re-popularized as “shifted symplectic structures”) in
Pavol Ševera, Some title containing the words "homotopy" and "symplectic", e.g. this one, based on a talk at “Poisson 2000”, CIRM Marseille, June 2000; (arXiv:0105080)
Dmitry Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids in Quantization, Poisson Brackets and Beyond , Theodore Voronov (ed.), Contemp. Math., Vol. 315, Amer. Math. Soc., Providence, RI, 2002 (arXiv)
The globally defined AKSZ action functionals obtained this way were shown in
to be a special case of the higher Lie integration process of
these were shown to be the infinitesimal version of the symmetries of prequantum n-bundles as they appear in local prequantum field theory, in higher generalization of how the Poisson bracket is the Lie algebra of the quantomorphism group.
These also encode a homotopy refinement of the Dickey bracket on Noether conserved currents which for Green-Schwarz sigma models reduces to Lie -algebras of BPS charges which refine super Lie algebras such as the M-theory super Lie algebra:
Comprehesive survey and exposition of this situation is in