# Generators of Lorentz group (algebra and action on spacetime)

My question is about the generators of the Lorentz group: signature $$(-,+,+,+)$$. I have found the well known Lorentz generators (intended as elements of its algebra evaluated in the identity element of the group)

Boosts: $$\begin{equation*} K_{1} =\begin{pmatrix} 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{pmatrix} \ \ \ K_{2} =\begin{pmatrix} 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{pmatrix} \ \ K_{3} =\begin{pmatrix} 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 \end{pmatrix} \end{equation*}$$

Rotations: $$\begin{equation*} J_{1} =\begin{pmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1\\ 0 & 0 & 1 & 0 \end{pmatrix} \ \ \ J_{2} =\begin{pmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \end{pmatrix} \ \ J_{3} =\begin{pmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 \end{pmatrix} \end{equation*}$$ with the following commutation relations $$\begin{equation*} [ J_{i} ,J_{j}] =\epsilon _{ijk} J_{k} \ \ \ \ [ K_{i} ,K_{j}] =-\epsilon _{ijk} J_{k} \ \ \ \ [ J_{i} ,K_{j}] =\epsilon _{ijk} K_{k} \end{equation*}$$

Then I want to find the generators of the action of the Lorentz group on spacetime, the induced vector fields, defined as $$$$V^\sharp |_x=\frac{\operatorname{d}}{\operatorname{d}t}\exp (tV)x \Bigl|_{t=0}$$$$ where $$V$$ is any of the previus generators, $$x$$ is a point of the spacetime and $$t$$ is a generic parameter.

For the Lorentz group I found the following generators of the action $$\begin{gather*} J^{\sharp }_{i} =\epsilon _{ijk} x^{j} \partial _{k} \ \Rightarrow \ J^{\sharp }_{1} =x^{2} \partial _{3} -x^{3} \partial _{2} \ \ \ \ J^{\sharp }_{2} =x^{3} \partial _{1} -x^{1} \partial _{3} \ \ \ \ J^{\sharp }_{3} =x^{1} \partial _{2} -x^{2} \partial _{1}\\ K^{\sharp }_{i} =x^{i} \partial _{0} +x^{0} \partial _{i} \ \Rightarrow \ K^{\sharp }_{1} =x^{1} \partial _{0} +x^{0} \partial _{1} \ \ \ \ K^{\sharp }_{2} =x^{2} \partial _{0} +x^{0} \partial _{2} \ \ \ \ K^{\sharp }_{3} =x^{3} \partial _{0} +x^{0} \partial _{3} \end{gather*}$$

My problem is that these generators don't have the same commutation relations of the generators of the group. For example $$[J_1^\sharp,J_2^\sharp]=-J_3^\sharp$$.

What am I wrong?

Suppose that $$G$$ is a Lie group, $$M$$ is a manifold and $$\lambda:G\times M\rightarrow M$$ is a smooth left action. There is then an induced mapping $$\lambda_\ast:\mathfrak g\rightarrow\mathfrak X(M)$$ of the Lie algebra $$\mathfrak g$$ into the Lie algebra of vector fields $$\mathfrak X(M)$$ that is an antihomomorphism, i. e. $$[\lambda_\ast X,\lambda_\ast Y]=-\lambda_\ast[X,Y]$$ for any $$X,Y\in\mathfrak g$$.
By contrast, for a right action, this relation is a genuine Lie algebra homomorphism. It is easy to turn a left action into a right action - use every element's inverse, i.e. $$\rho_gx=\lambda_{g^{-1}}x$$, then $$\rho$$ is a right action.
So using OP's notation, if $$V^\sharp$$ is defined as $$V^\sharp_x=\frac{d}{dt}\exp(-tV)x|_{t=0},$$ then the commutation relations will come out as intended.