∞-Lie theory (higher geometry)
Formal Lie groupoids
The Poincaré Lie algebra is the semidirect product of the special orthogonal Lie algebra and the abelian translation Lie algebra .
The corresponding Lie group is the Poincaré group.
The Chevalley-Eilenberg algebra is generated from and . For the standard basis of we write and for these generators. With the components of the Minkowski metric we write
In terms of this the CE-differential that defines the Lie algebra structure is
We discuss some elements in the Lie algebra cohomology of .
The canonical degree-3 -cocycle is
The volume cocycle is the volume form
Invariant polynomials and Chern-Simons elements
With the basis elements as above, denote the shifted generators of the Weil algebra by and , respectively.
We have the Bianchi identity
The element is an invariant polynomial. A Chern-Simons element for it is . So this transgresses to the trivial cocycle.
Another invariant polynomial is . This is the Killing form of . Accordingly, it transgresses to a multiple of .
This is the first in an infinite series of Pontryagin invariant polynomials
There is also an infinite series of mixed invariant polynomials
Chern-Simons elements for these are
Lie algebra valued forms
A Lie algebra-valued form with values in
The curvature 2-form consists of
If the torsion vanishes, then is a Levi-Civita connection for the metric defined by .
The volume form is the image of the volume cocycle
If the torsion vanishes, this is indeed a closed form.