There are two different concepts called Weil algebra. This entry is about the notion of Weil algebra in Lie theory. For the notion in infinitesimal geometry see infinitesimally thickened point/Artin algebra.
Formal Lie groupoids
The notion of Weil algebra is ordinarily defined for a Lie algebra . It may be understood as the Chevalley-Eilenberg algebra of the tangent Lie 2-algebra or of , generalizing the notion of tangent Lie algebroid from a 0-truncated Lie algebroid (a smooth manifold) to the one-obeject Lie algebroid .
Generally, for every Lie-∞-algebroid one may define the corresponding tangent Lie--algebroid , whose Chevalley-Eilenberg algebra may be called the Weil algebra of :
Weil algebra of a Lie algebra
Let be a finite-dimensional Lie algebra. The Weil algebra of is
the graded Grassmann algebra generated from the dual vector space together with another copy of shifted in degree
equipped with a derivation that makes this a dg-algebra, defined by the fact that on it acts as the differential of the Chevalley-Eilenberg algebra of plus the degree shift morphism .
This Weil algebra has trivial cohomology everywhere (except in degree 0 of course) and sits in a sequence
with the Chevalley-Eilenberg algebra of and its algebra of invariant polynomials on . This may be understood as a model for the sequence of algebras of differential forms on the universal G-bundle
As such, the Weil algebra plays a crucial role in the study of the Lie algebra cohomology of .
We first consider Weil algebras of L-∞ algebras, then more generally of L-∞ algebroids.
We use the notation and grading conventions that are described in detail at Chevalley-Eilenberg algebra.
Let be an L-∞ algebra of finite type. By our grading conventions this means that the graded vector space obtained by degreewise dualization is in non-negative degree, and is its shift up into positive degree.
A quick abstract way to characterize the Weil algebra of is as follows. Notice that there is a free functor/forgetful functor adjunction
between the category dgAlg of dg-algebras and the category of -graded vector spaces (all over some fixed field). Notice that a free object is unique up to isomorphism .
The Weil algebra is the unique representative of the free dg-algebra on for which the projection of graded vector spaces extended to a dg-algebra homomorphism
We discuss below in the Properties section that this is equivalent to the following component-wise definition
The Weil algebra is the semi-free dga whose underlying graded-commutative algebra is the exterior algebra
on and a shifted copy of , and whose differential is the sum
of two graded derivations of degree +1 defined by
acts by degree shift on elements in and by 0 on elements of ;
acts on unshifted elements in as the differential of the Chevalley-Eilenberg algebra of and is extended uniquely to shifted generators by graded-commutattivity
for all .
Where the Chevalley-Eilenberg algebra of an L-∞ algebra has in degree 0 the ground field, that of an L-∞ algebroid has more generally an algebra over a Lawvere theory. For L-∞ algebroids over smooth manifolds this is the algebra of smooth functions on a manifolds, regarded as a smooth algebra (-ring).
So let be a Fermat theory. Write for the corresponding category of algebra. There is a free functor/forgetful functor adjunction
to the category CRing of commutative Rings.
We need the facts that
with values in the -module of -Kähler differentials.
See the corresponding entries for more details. The second point means that for any -derivation on , there is a unique -module homomorphism
such that the diagram
Let now be an L-∞ algebroid with Chevalley-Eilenberg algebra considered as the following data;
a graded commutative semifree dga over the ground field;
the structure of a -algebra on the associative algebra (over the ground field)
such that is a derivation of -algebra modules.
By semi-freeness there exists a -graded vector space and an isomorphism
The Weil algebra of the -algebroid is the Chevalley-Eilenberg algebra of the -algebroid defined as follows
the -algebra in degree 0 is the same as that of ;
the underlying graded algebra is the exterior algebra on and a shifted copy as well as one copy of the Kähler differential module in lowest degree (though of as the shifted copy of itself)
the differential is the sum
of two degree +1 graded derivations, where and are defined on as above for -algebras and on itself vanishes and acts as the universal derivation
The main point of the definition is that the differential restricted to the original (unshifted) generators is the original differential plus the shift:
By solving the condition and using that this already fixes uniquely the differential . To see this we only need to show that the value of on a generator is completely determined by . One computes:
This implies the following universal freeness property:
Let be an -algebra. Morphisms of -algebras are in natural bijection to morphisms of graded vector spaces .
Forgetting the differential, is the free graded-commutative algebra generated by (a shifted copy of) and . Therefore,
Projecting down to , one obtains a natural map
which is a bijection.
To prove injectivity, we just have to show that the restriction of a dgca morphism to determines the restriction of to . One has, for any ,
Since lies in the sub-gca of generated by , the element , and therefore , is determined by .
Next we show surjectivity, i.e. that every morphism of graded vector spaces can be extended to a dgca morphism . Denote by the extension of to a graded commutative algebra morphism, and let be the graded vector space morphism defined by
for any . The graded vector space morphism extends to a commutative graded algebra , whose restriction to is . We want to show that is actually a dgca morphism. We only need to test commutativity with the differentials on generators and . We have
which in particular implies that , and
Since , we obtain
For the Chevalley-Eilenberg algebra of , the inclusion induces a canonical surjective dgca morphism . This is the identity on the unshifted generators, and 0 on the shifted generators.
For the de Rham complex of a smooth manifold , we have that
is the collection of total degree 1 differential forms with values in the -Lie algebra .
A morphism of
sends the unshifted generators to differential forms , which one thinks of as local connection forms, and sends the shifted generators to their curvature. The respect for the differential on the shifted generators is the Bianchi identity on these curvatures.
A morphism encodes a collection of flat -algebra valued forms precisely if it factors by the canonical morphism from above through the Chevalley-Eilenberg algebra of .
The freeness property of the Weil algebra can be made more explicit by exhibiting a concrete isomorphism to the free dg-algebra on .
The canonical free dg-algebra on is
where the differential is on the unshifted generators the shift isomorphism extended as a derivation and vanishes on the shifted generators
Or in other words, if is the -Lie algebra whose underlying graded vector space is that of , but all whose brackets vanish, then
Notice the evident
To see this, let be the degree down-shift isomorphism extended as a graded derivation of degree -1, then
and hence for any such that we have .
Given , there is an isomorphism of dg-algebras
It is clear that is a dg-algebra homomorphism. The inverse dg-algebra morphism is given on generators by
Note that the isomorphism is precisely the dgca isomorphism induced between and by the identity of as a graded vector spaces morphism .
The cochain cohomology of the Weil algebra of an -algebra is trivial.
As the CE-algebra of the -algebra of inner derivations
By the discussion at ∞-Lie algebra and Chevalley-Eilenberg algebra, we may identify the full subcategory of the opposite category dgAlg on commutative semi-free dgas in non-negative degree with that of ∞-Lie algebras/∞-Lie algebroids.
That means that the Weil algebra of some L-∞ algebra is the Chevalley-Eilenberg algebra of another -Lie algebra.
For any -Lie algebra write for the -Lie algebra whose CE-algebra is :
In the following we discuss these inner automorphism -Lie algebras in more detail. (See section 6 of (SSSI)).
For an ordinary Lie algebra
For an ordinary Lie algebra the inner derivation Lie 2-algebra is the strict Lie 2-algebra given by the dg-Lie algebra
elements in degree -1 are the elements , thought of as inner degree-(-1) derivations
given by contraction with ;
elements in degree 0 are the derivations of degree 0 that are of the form
the differential is the commutator of derivations with the differential ;
the bracket is the graded commutator of derivations.
Equivalently this is identified with the differential crossed module with the action being the adjoint action of on itself.
One checks that for all we have in the brackets
and of course
These identities are known as Cartan calculus. In this context is called a Lie derivative.
In this sense one may understand for general -Lie algebras as providing an -version of Cartan calculus.
Relation to other concepts
-Lie algebra valued differential forms
For an ∞-Lie algebra, a smooth manifold, an ∞-Lie algebra valued differential form is a morphism
of dg-algebras, from the Weil algebra into the de Rham complex of .
The image of the unshifted generators are the forms themselves, the image of the shifted generators are the corresponding curvatures. The respect for the differential on the shifted generators are the Bianchi identity on the curvatures.
Precisely if the curvatures vanish does the morphism factor through the Chevalley-Eilenberg algebra .
Invariant polynomials and Chern-Simons elements
A cocycle in the ∞-Lie algebra cohomology of the ∞-Lie algebra is a closed element in the Chevalley-Eilenberg algebra .
An invariant polynomial on is a closed element in the Weil algebra , subject to the additional condition that it its entirely in the shifted copy of , .
For an element of the -Lie algebra, let
the evident operation of contraction with
extended as a graded derivation. Then the Lie derivative
encodes the coadjoint action of on . By the above definition of an invariant polynomial , we have
Since the cohomology of is trivial, there is necessarily for each invariant polynomial an element such that
This is the Chern-Simons element of the invariant polynomial. Notice, crucially, that this is ingeneral not restricted to the shifted part Its restriction
to the unshifted copy, hence to the Chevalley-Eilenberg algebra, is the cocycle that is in transgression with .
a collection of -valued differential forms (as above) and an invariant polynomial, the composite
is the corresponding curvature characteristic form, a closed -form on . For the corresponding Chern-Simons element we have that is the corresponding Chern-Simons form on .
Weil algebra of a Lie algebra
Let be a finite dimensional Lie algebra. This Lie algebra regarded as a Lie algebroid has as base manifold the point, . Its algebra of functions is accordingly the ground field, and the algebra is just a Grassmann algebra.
The Chevalley-Eilenberg algebra is
where the differential acts on the elements of in degree 1 by the linear dual of the Lie bracket.
The corresponding Weil algebra is obtained by adding another copy of in degree 2
where with the degree shift isomorphism, the differential acts as
For illustration, we spell this out in a basis.
Let be a basis for the underlying vector space of and let be the corresponding structure constants of the Lie bracket
Then the Chevalley-Eilenberg algebra is generated on generators of degree 1, on which the differential acts as
The Weil algebra in turn is generated from these generators in degree 1 and generators in degree 2, with differential given by
Weil algebra of a 0-Lie algebroid
A 0-truncated Lie algebroid is one for which the chain complex of modules over the -algebra in degree 0 vanishes:
For instance for =CartSp the theory of smooth algebras, any smooth manifold regarded as an L-∞ algebroid is a 0-Lie algebroid with the smooth algebra of smooth functions on .
The Weil algebra of a 0-Lie algebroid is the Kähler de Rham complex of :
This Weil algebra is the Chevalley-Eilenberg algebra of the tangent Lie algebroid of , which is the de Rham algebra of :
Among the original references on Weil algebras for ordinary Lie algebras is
- Henri Cartan, Cohomologie réelle d’un espace fibré principal diffrentielle I, II, Séminaire Henri Cartan, 1949/50, pp. 19-01 – 19-10 and 20-01 – 20-11, CBRM, (1950).
- Henri Cartan, Notions d’algébre différentielle; application aux groupes de Lie et aux variétés ou opère un groupe de Lie , Colloque de topologie (espaces fibrs), Bruxelles, (1950), pp. 15–27.
This also explains the use of the Weil algebra in the calculation of the equivariant de Rham cohomology of manifolds acted on by a compact group. These papers are reprinted, explained and put in a modern context in the book
A clasical textbook account of standard material is in chapter VI, vol III of
Some remarks on the notation there as compared to ours: our is their on p. 226 (vol III). Their is our . Their is our (/ denoting the representation)..
Some related material is also in
- Victor Guillemin, Shlomo Sternberg, Supersymmetry and equivariant de Rham theory, Springer, (1999).
The (obvious but conceptually important) observation that Lie algebra-valued 1-forms regarded as morphisms of graded vector spaces are equivalently morphisms of dg-algebras out of the Weil algebra and that one may think of as the identity as the universal -connection appears in early articles for instance highlighted on p. 15 of
- Franz W. Kamber; Philippe Tondeur, Semisimplicial Weil algebras and characteristic classes for foliated bundles in Cech cohomology , Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 1, pp. 283–294. Amer. Math. Soc., Providence, R.I., (1975).
A survey of Weil algebras for Lie algebras is also available at
Weil algebra for L-infinity algebras and their role in defining invariant polynomials and Chern-Simons elements on -Lie algebras from L-infinity algebra cocycle are considered in