Lie Groups/Algebras and Structures on a Vector Space

The concept of ``a structure on a vector space'' is a huge unifying
concept, which organizes a great deal of otherwise erratic info.

The key insight, and the one that makes this tractible, is Klein's
(in context of geometry; perhaps someone else had it before):
understand a ``structure on $X$'' by the symmetries that preserve it,
viz, the submonoid of $\End X$ (often the subgroup of $\Aut X$).
Indeed, if two structures have the same symmetries, they are in some
sense ``the same''.

Thus we've reduced the hopelessly intractible problem of ``any
possible structure you can imagine on $V$'' to:
Submonoids or subgroups of $\End V$ or $\Aut V$.

In turn, we can break this up into:
- what -abstract- groups can be realized?
- -how- can they be realized/embedded?
In other words, representation theory.

As these groups are manifolds, we can distinguish them into
- continuous
- discrete

For the continuous groups, the abstract answer is:
- the semisimple Lie Algebras (and groups)
  (plus the solveable part and extensions)
  which are the 4 families:
  [emphasize the difference between odd and even orthogonal]
    $\sl_{n+1}, \so_{2n}, \so_{2n+1}, \sp_{2n}$
    (which correspond to $\bR, \bC, \bH$)
  and the 5 exceptions:
    $\fe_6, \fe_7, \fe_8, \ff_4, \fg_2$
    (which correspond to $\bO$)
- The real groups is a bit more work
- the embedded answer is: the representation theory of ssla,
  which is elegant but a rather long story.
  The geometric meaning is often forgotten:
    okay, what geometric structure does this representation correspond
    to?

For the discrete groups, the answer is:
- every finite group has a faithful representation,
  b/c it's a subgroup of a $S_n$
  This is the representation theory of finite groups,
  which is another very beautiful and important story.
  Again, one should ask: what geometric structure does this
  representation correspond to?
    (I virtually never know the answer; for cyclic groups the answer
     is ``polygons/stars'')
  
- there are certain -natural- finite group actions,
  namely:
  - $\pi_0$ of the classical groups
    (careful: $\pi_0$ doesn't act itself;
     we're talking about a quotient action,
     or rather, the structure corresponding to $G_0$)
  - the Weyl group
    ...which leads to Frobenius-Schur duality
      ($S_n$ and $U_n$), etc.
    [and the study of the Coexeter groups]

Other natural groups show up when studying Lie Groups,
like the Bruhat group (stabilize flag) and tori (stabilize axes) etc.

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Concretely, what are some things to preserve?
- naively
  + an element of $V$,
    or some vector/set functor of $V$
  (e.g.: a bilinear form on it; a subset (or ordered subset) of it;
    etc.; one can give such descriptions for (most?) of the below,
    though this isn't the way to go looking for 'em)

- geometrically meaningfully:
continuous:
 + $\sl_{n+1}$: volume form
   (real: volume \& orientation)
   (complex: volume \& ``orientation'')
   non-zero element of $\Alt^n V$
 + $\so_{2n}$ \& $\so_{2n+1}$, orthogonal/Riemannian (even/odd)
   element of $\Sym^2 V \cap \Iso(V,V^*) < V^{\otimes 2}$
 + $\sp_{2n}$ symplectic
   element of $\Alt^2 V \cap \Iso(V,V^*) < V^{\otimes 2}$
  ...and that's it for -abstract- -complex- structures,
  other than the exceptional groups
    (which are related to the one interesting trilinear form)
For the reals, there's a bit more:
 + $O(p,q)$ (pseudo-Riemannian);
 + $U(n)$ and $SU(n)$
 + almost-complex/almost-quaternionic structure
 + $\Spin(n)$; $\Spin^c$?; etc.
 (the various $\PGL$ etc., and the intermediate $\PSL$;
  yeah, I'm going for completeness)

discrete: orientation, direction of time (for $O(p,q)$)
  [there aren't many here b/c the groups are often connected,
   and have few components even if not]
  One can also look at structure coming from $\pi_1$
  (Spin, and something for $\PSL$; and something from $E_8$?)
  ...but once you get to $\pi_3$ (there's no $\pi_2$) you get into
  infinite-dimensional groups
  (and $\pi_{-1} O$ corresponds to $\pi_0 BO = \pi_7 O$ = dimension
   and you get Bott periodicity)

BTW, don't forget $\SL^{\pm}$ and $\Sp^{\pm}$: they're natural analogs
of $O(n)$, and do show up (in orientation-reversing maps of surfaces,
for instance).

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Wow!
The upshot is: thinking in terms of ``structure on a vector space''
organizes an -assload- of the topic.
