∞-Lie theory (higher geometry)
Formal Lie groupoids
The Poincaré Lie algebra is the Lie algebra of the isometry group of Minkowski spacetime: the Poincaré group. This happens to be the semidirect product of the special orthogonal Lie algebra with the the abelian translation Lie algebra .
For , write for Minkowski spacetime, regarded as the inner product space whose underlying vector space is and equipped with the bilinear form given in the canonical linear basis of by
The Poincaré group is the isometry group of this inner product space. The Poincaré Lie algebra is the Lie algebra of this Lie group (its Lie differentiation)
For the canonical linear basis of , and for the corresponding canonical basis of , then the Lie bracket in is given as follows:
Since Lie differentiation sees only the connected component of a Lie group, and does not distinguish betwee a Lie group and any of its discrete covering spaces, we may equivalently consider the Lie algebra of the spin group (the double cover of the proper orthochronous Lorentz group) and its action on .
By the discussion at spin group, the Lie algebra of is the Lie algebra spanned by the Clifford algebra bivectors
and its action on itself as well as on the vectors, identified with single Clifford generators
is given by forming commutators in the Clifford algebra:
Via the Clifford relation
this yields the claim.
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.