Pensieve header: Verifying the Duflo commutation relations, as communicated by Dylan in 1999.

Date: Sun, 23 May 1999 17:24:53 +0200 (MET DST)
From: Dylan Thurston <Dylan.Thurston@math.unige.ch>
To: Dror Bar-Natan <drorbn@math.huji.ac.il>
Subject: Oh, yeah...

Duflo had another possibly useful comment: he said that there is a simpler
(non-trivial) algebra than sl_2, given by the relations (more or less)

        [x,y] = z
        [x,t] = x
        [y,t] = -y
        [t,z] = [x,z] = [y,z] = 0

I hope I got that right; it should be the Lie algebra of matrices

        0 x z
        0 t y
        0 0 0

He says that one can write down the exponential explicitly, that
commutators very quickly go to zero, etc.  Apparently it plays the same
role for reducible lie algebras that sl_2 plays for semi-simple ones.
It’s worth thinking about.

Best,
        Dylan

x=(({{0, 1, 0}, {0, 0, 0}, {0, 0, 0}})); y=(({{0, 0, 0}, {0, 0, 1}, {0, 0, 0}})); z=(({{0, 0, 1}, {0, 0, 0}, {0, 0, 0}})); t=(({{0, 0, 0}, {0, 1, 0}, {0, 0, 0}}));  o=(({{0, 0, 0}, {0, 0, 0}, {0, 0, 0}})); 
{x.y-y.x==z,x.t-t.x==x,y.t-t.y==-y,t.z-z.t==o,x.z-z.x==o,y.z-z.y==o}

MatrixExp[α x+β y+γ z+δ t] // MatrixForm

MatrixExp[α x].MatrixExp[β y].MatrixExp[γ z].MatrixExp[δ t] // MatrixForm

Adding an “Euler” element:

d=(({{0, 0, 0}, {0, 0, 0}, {0, 0, 1}})); {d.x-x.d==o, d.t-t.d==o, d.y-y.d==-y,d.z-z.d==-z}

MatrixForm /@ {M=δ α x+δ β y+γ z+δ t, MatrixExp[M]}

MatrixExp[(Subscript[α, 1]) x+(Subscript[β, 1]) y+(Subscript[γ, 1]) z+(Subscript[δ, 1]) t].MatrixExp[(Subscript[α, 2]) x+(Subscript[β, 2]) y+(Subscript[γ, 2]) z+(Subscript[δ, 2]) t]  // MatrixForm

MatrixExp[(Subscript[α, 1]) x+(Subscript[β, 1]) y+(Subscript[γ, 1]) z+(Subscript[δ, 1]) t].MatrixExp[(Subscript[α, 2]) x+(Subscript[β, 2]) y+(Subscript[γ, 2]) z+(Subscript[δ, 2]) t]  //MatrixLog // PowerExpand // FullSimplify// MatrixForm

MatrixExp[(Subscript[δ, 1])(Subscript[α, 1]) x+(Subscript[δ, 1])(Subscript[β, 1]) y+(Subscript[γ, 1]) z+(Subscript[δ, 1]) t].MatrixExp[(Subscript[δ, 2])(Subscript[α, 2]) x+(Subscript[δ, 2])(Subscript[β, 2]) y+(Subscript[γ, 2]) z+(Subscript[δ, 2]) t]  //MatrixLog // PowerExpand // Simplify// MatrixForm

MatrixExp[(Subscript[α, 1]) x+(Subscript[β, 1]) y+(Subscript[γ, 1]) z].MatrixExp[(Subscript[α, 2]) x+(Subscript[β, 2]) y+(Subscript[γ, 2]) z]  //MatrixLog // PowerExpand // FullSimplify// MatrixForm

MatrixExp[(Subscript[α, 1]) x+(Subscript[β, 1]) y].MatrixExp[(Subscript[α, 2]) x+(Subscript[β, 2]) y]  //MatrixLog // PowerExpand // FullSimplify// MatrixForm

MatrixExp[(Subscript[α, 1]) x+(Subscript[δ, 1]) t].MatrixExp[(Subscript[α, 2]) x+(Subscript[δ, 2]) t]  //MatrixLog // PowerExpand // FullSimplify// MatrixForm

MatrixExp[(Subscript[δ, 1])(Subscript[α, 1]) x+(Subscript[δ, 1]) t].MatrixExp[(Subscript[δ, 2])(Subscript[α, 2]) x+(Subscript[δ, 2]) t]  //MatrixLog // PowerExpand // FullSimplify// MatrixForm