What is the point of the sympletic lie algebra

Question

I have taken a lie algebra class this semester, and have stumbled upon a particularly unmotivated lie algebra, the sympleitc lie algebra. So my question is quite simple, why is this an interesting example of a lie algebra? What desirable property does it have? Or in what interesting consequence does it appear?

Answer 1

I assume you are already familiar with the (special) orthogonal Lie algebra which comprises the endomorphisms that are skew for a symmetric bilinear form. A natural change we can make is swap symmetric for skew-symmetric and then we get the symplectic Lie algebra as those endomorphisms which are skew for a symplectic (i.e. nondegenerate, skew-symmetric, bilinear) form.

Being skew for these forms (i.e. $(Xv,w) + (v,Xw) = 0$) being the natural Lie algebra equivalent of the forms being invariant for the corresponding Lie groups $(gv,gw) = (v,w)$

Of course we could instead change to sequilinear forms and get the special unitary Lie algebra. Hopefully, put like this it is easy to see that it is quite natural that these should all live together in the family of simple Lie algebras.

As to why symplectic Lie algebra/groups are worth studying in their own right we could turn to the fairly sizable field that is symplectic geometry which has important applications in physics. A non-physics application is that the coadjoint orbits (i.e. orbits in the dual of a Lie algebra under the action of its Lie group) carry a natural symplectic structure.