Lorentz algebra and group question with regards to operator representaion of $M^{\mu\nu}$

1) Likely an Einstein summation confusion. Consider Lorentz transformation's defined in the following matter:

I aim to consider the product $$L^0{}_0(\Lambda_1\Lambda_2).$$ Consider the following notation $$L^\mu{}_\nu(\Lambda_i) = L_i{}^\mu{}_\nu$$. How then, does $$L^0{}_0(\Lambda_1\Lambda_2) = L_1{}^0{}_\mu L_2{}^\mu{}_0?$$

2) To write the J and K generators of the Lorentz group in a compact way, one can write $$(M^{lm})^j{}_k=i (g^ {lj}g^m{}_k - g^{mj}g^l{}_k)$$ where on the left hand side, it is helpful to think of l and m as indices/labels, and the j and k as rows/columns for the whole matrix. (Apparently) One can write this in an operator representation as $$M^{\mu\nu} = i(x^\mu \partial^\nu-x^\nu\partial^\mu).$$

a) where does this come from?

b) why and how is it used? What is it operating on, $$x^\mu?$$

c) how does $$\partial^\nu x^\sigma= \frac{\partial}{\partial x_\nu} x^\sigma$$ equal $$g^{\nu\sigma}$$?

• Regarding c): Work out the 16 values of $\frac{\partial}{\partial x_\nu}x^\sigma$. – G. Smith Oct 8 at 15:43