so consider the operad for "lie algebras where all of the tetralinear operations are trivial"... so the underlying schur functor has terms corresponding to the trivial irrep of 1!, the sign rep of 2!, and the 2d irrep of 3! ... or something like that...

so the free algebra of this operad on a 2d vector space v is... the direct sum of v and its exterior square and... ??another copy og v? ...

and the "holomorph" of this lie group should be contained in g2?? in a hopefully obvious way?? ... visible in the root system... ??

??some confusion here about... holomorph... ??... automorphisms from gl(2), vs inner automorphisms, vs translations... left vs right ... ??....

consider also the b2 case here... heisenberg alg of a 2d vector space...




nothing, element, frame

1 1 1
1 2 3
1 3 6

"categorified gram-schmidt"

(so what about "categorified iwasawa decomposition" or something?? ... and so forth...)

1 -1 1
0 1 -2
0 0 1

1 -2 1
-2 4 -2
1 -2 1

x(x-1)(x-2)/6 + x(x-1)/2 + x

... ??...