∞-Lie theory (higher geometry)
Formal Lie groupoids
The Chevalley-Eilenberg algebra of a Lie algebra is a differential graded algebra of elements dual to whose differential encodes the Lie bracket on .
The cochain cohomology of the underlying cochain complex is the Lie algebra cohomology of .
This generalizes to a notion of Chevalley-Eilenberg algebra for an L-∞-algebra, a Lie algebroid and generally an ∞-Lie algebroid.
This differential-graded subject is somewhat notorious for a plethora of equivalent but different conventions on gradings and signs.
For the following we adopt the convention that for an -graded vector space we write
for the free graded-commutative algebra on the graded vector space obtained by shifting up in degree by one.
Here the elements in the th term in parenthesis are in degree .
A plain vector space, such as the dual of the vector space underlying a Lie algebra, we regard here as a -graded vector space in degree 0. For such, is the ordinary Grassmann algebra over , where elements of are generators of degree 1.
Of Lie algebras
The Chevalley-Eilenberg algebra of a finite dimensional Lie algebra is the semifree graded-commutative dg-algebra whose underlying graded algebra is the Grassmann algebra
(with the th skew-symmetrized power in degree )
and whose differential (of degree +1) is on the dual of the Lie bracket
extended uniquely as a graded derivation on .
That this differential indeed squares to 0, , is precisely the fact that the Lie bracket satisfies the Jacobi identity.
If we choose a dual basis of and let be the structure constants of the Lie bracket in that basis, then the action of the differential on the basis generators is
where here and in the following a sum over repeated indices is implicit.
This has a more or less evident generalization to infinite-dimensional Lie algebras.,
One observes that for a vector space, the graded-commutative dg-algebra structures on are precisely in bijection with Lie algebra structures on : the dual of the restriction of to defines a skew-linear bracket and the condition holds if and only if that bracket satisfies the Jacobi identity.
Moreover, morphisms if Lie algebras are precisely in bijection with morphisms of dg-algebras . And the -construction is functorial.
Therefore, if we write for the category whose objects are semifree dgas on generators in degree 1, we find that the construction of CE-algebras from Lie algebras constitutes a canonical equivalence of categories
where on the right we have the opposite category.
This says that in a sense the Chevalley-Eilenberg algebra is just another way of looking at (finite dimensional) Lie algebras.
There is an analogous statement not involving the dualization: Lie algebra structures on are also in bijection with the structure of a differential graded coalgebra on the free graded-co-commutative coalgebra on with a derivation of degree -1 squaring to 0.
The relation between the differentials is simply dualization
where for each we have
The equivalence between Lie algebras and differential graded algebras/coalgebras discussed above suggests a grand generalization by simply generalizing the Grassmann algebra over a vector space to the Grassmann algebra over a graded vector space.
If is a graded vector space, then a differential of degree -1 squaring to 0 on is precisely the same as equipping with the structure of an L-∞ algebra.
Dually, this corresponds to a general semifree dga
This we may usefully think of as the Chevalley-Eilenberg algebra of the -algebra .
So every commutative semifree dga (degreewise finite-dimensional) is the Chevaley-Eilenberg algebra of some L-∞ algebra of finite type.
This means that many constructions involving dg-algebras are secretly about ∞-Lie theory. For instance the Sullivan construction in rational homotopy theory may be interpreted in terms of Lie integration of -algebras.
Of Lie algebroids
For a Lie algebroid given as
a vector bundle
with anchor map
the corresponding Chevalley-Eilenberg algebra is
where now the tensor products and dualization is over the ring of smooth functions on the base space (with values in the real numbers). The differential is given by the formula
for all and , where denotes the set of -shuffles and the signature of the corresponding permutation.
For the point we have that is a Lie algebra and this definition reproduces the above definition of the CE-algebra of a Lie algebra (possibly up to an irrelevant global sign).
Of -Lie algebroids
See ∞-Lie algebroid.
Of abelian Lie -algebras
The CE-algebra of the Lie algebra of the circle group is the graded-commutative dg-algebra on a single generator in degree 1 with vanishing differential.
More generally, the -algebra is the one whose CE algebra is the commutative dg-algebra with a single generator in degree and vanishing differential.
The CE-algebra of has two generators in degree one and differential
Of the tangent Lie algebroid
For a smooth manifold and its tangent Lie algebroid, the corresponding CE-algebra is the de Rham algebra of .
For vector fields and a differential form of degree , the formula for the CE-differential
is indeed that for the de Rham differential.
Of the string Lie 2-algebra
For a semisimple Lie algebra with binary invariant polynomial – the Killing form – , the CE-algebra of the string Lie 2-algebra is
where the differential restricted to is while on the new generator spanning is it
For a Lie algebra, the CE-algebra of the Lie 2-algebra given by the differential crossed module is the Weil algebra of
Lie algebra cohomology
Lie algebra cohomology of a -Lie algebra with coefficients in the left -module is defined as . It can be computed as (a similar story is for Lie algebra homology) where is the Chevalley-Eilenberg chain complex. If is finite-dimensional over a field then is the underlying complex of the Chevalley-Eilenberg algebra, i.e. the Chevalley-Eilenberg cochain complex with trivial coefficients.
A cocycle in degree n of the Lie algebra cohomology of a Lie algebra with values in the trivial module is a morphism of L-∞ algebras
In terms of CE-algebras this is a dg-algebra morphism
Since by the above example the dg-algebra on he right has a single generator in degree and vanishing differential, such a morphism is precisely the same thing as a degree -element in , i.e. an element which is closed under the CE-differential
This is what one often sees as the definition of a cocycle in Lie algebra cohomology. However, from the general point of view of cohomology, it is better to think of the cocycle equivalently as the morphism .
In physics, the Chevalley-Eilenberg algebra of the action of a Lie algebra or L-∞ algebra of a gauge group on space of fields is called the BRST complex.
In this context
the generators in in degree 0 are called fields;
the generators in degree are called ghosts;
the generators in degree are called ghosts of ghosts;
If is itself a chain complex, then this is called a BV-BRST complex
An elementary introduction for CE-algebras of Lie algebras is at the beginning of
- J. A. de Azcarraga, J. M. Izquierdo, J. C. Perez Bueno, An introduction to some novel applications of Lie algebra cohomology and physics (arXiv)
More details are in section 6.7 of
- J. A. de Azcárraga, José M. Izquierdo, Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics , Cambridge monographs of mathematical physics, (1995)
See also almost any text on Lie algebra cohomology (see the list of references there).