# Lie Bracket on Lie Algebra vs. Lie derivative on corresponding group orbit tangent vector field

Suppose I have a matrix group, for example the 2-d affine group and two elements of Lie algebra, A and B, expressed as 3x3 matrices. The Lie bracket is [A,B]=AB-BA. Suppose on the other hand I generate vector fields in the underlying 2-d x-y plane, Va(p) = Ap and Vb(p) = Bp where $p = [x,y,1]^t$. If I apply the general Lie bracket on these vector fields,

$[Va, Vb] = \hat{x}(Va_x\frac{\partial Vb_x}{\partial x}+Va_y\frac{\partial Vb_x}{\partial y}-Vb_x\frac{\partial Va_x}{\partial x}-Vb_y\frac{\partial Va_x}{\partial y})+\hat{y}(Va_x\frac{\partial Vb_y}{\partial x}+Va_y\frac{\partial Vb_y}{\partial y}-Vb_x\frac{\partial Va_y}{\partial x}-Vb_y\frac{\partial Va_y}{\partial y})$

I get the result that $[Va, Vb] = [B,A]p$, which is just the reverse order of what I expected. Where did I go wrong?

Thanks, JLM

• You did not get anything wrong. For a left action of a Lie group on a manifold, the map associating to an element of the Lie algebra the fundamental vector field is an anti-homomorphism, not a homomorphism. Hence $[\zeta_X,\zeta_Y]=\zeta_{-[X,Y]}=\zeta_{[Y,X]}$. – Andreas Cap Oct 1 '15 at 17:58
• Andreas, Thanks! What is a good reference that explains this relationship?. JLM – J.Mundy Oct 2 '15 at 11:58
• You can look at Section 6 of P. Michor's "Topics in differential geometry", which is available in PDF via his homepage. – Andreas Cap Oct 2 '15 at 12:25