When g is defined over a field of characteristic zero, g is semisimple if and only if every representation is completely reducible, that is for every invariant subspace of the representation there is an invariant complement (Weyl's theorem).
For a kinematic Heisenberg algebra H(m) of QM in m [3] dimensional space, usually given as (generally unbounded) operators CCR acting on a specifically constructed Hilbert space, there is, axiomatically, and more abstractly, a nilpotent Liealgebra of 2m+1 dimensions given by by the defining Commutation Relations (CRs).
In the case of the Heisenberg algebra, there is an interesting cross relationship within a q-p pair familiar from classical canonical mechanics stated as, momentum is the generator of spatial translations and position is the generator of momentum translations in the context of phase space.
Semisimplealgebras can be understood as constructed as a direct sum of simple Liealgebras, where the representation theory has the representations su(2) or so(3) playing a seminal role.
Feeds |
Contact