A Short Note on the Poincaré Algebra
May 22, 2009 – 7:03 pmAs physicists, we learn that the Poincaré algebra has two Casimirs, 
and 
, together describing the mass and spin of a particle. A standard question is then, “why does the algebra have two Casimirs?” and the standard answer is, “because it is a rank 2 algebra, for instance taking 
as the Cartan”. Well this seems wrong, since we can also take 
as the Cartan, which is of dimension 4.
It is a standard result that any Cartan subalgebra of a (complex) semisimple Lie algebra has the same size, so what’s going on?
The answer is simple: Poincaré isn’t a semisimple Lie algebra. Therefore we have to be careful about how to define the rank. First, let’s see why Poincaré isn’t semisimple.
Definition. If 
is a complex Lie algebra, then an ideal in is a complex subalgebra 
of such that, for all 
and 
, ![[X,H] \in h](/latexrender/pictures/ac35b9c3c2753689beabe9a005c524b4.png "[X,H] \in h")
.
The brackets of a Lie algebra can be thought of as a product of two elements in that algebra. Then, an ideal (as always), is a sort of ‘zero’, making anything it multiplies a member of itself (just like 
for any 
).
Definition. A complex Lie algebra is called simple if 
, and the only ideals in are and 
.
Definition. A complex Lie algebra is called semisimple if it’s (isomorphic to) a direct sum of simple Lie algebras.
We can now see why the Poincaré algebra isn’t simple. Translations, namely , form a basis for an ideal: Translations commute, and the commutator of a translation with a rotation 
is a sum of translations. It is also not semisimple, which I guess can be seen by considering the 
decomposition of the Lorentz subalgebra, then adding translations which will ‘link’ the two components.
The Cartan can be defined for non-semisimple algebras, and it turns out it is the Cartan of the largest semisimple subalgebra. In the case of Poincaré, the largest semisimple subalgebra is the Lorentz subalgebra. So we can still define the rank to be the size of the Cartan, with this more general definition. I don’t know if the relation between the number of Casimirs and the rank still holds in this case, but at least for the Poincaré algebra it does turn out correctly, since the rank of the Lorentz algebra is 2.
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One Response to “A Short Note on the Poincaré Algebra”
To me, and to generations of French kids, there is only one Casimir.
By Dan Elharrar on May 25, 2009