# Representations of $sl(2,C)$ as a real Lie algebra

Let $sl(2,C)_R$ be the decomplexification of the complex Lie algebra $sl(2,C)$. Therefore $sl(2,C)_R$ is a real Lie algebra with the same elements as $sl(2,C)$.

I am trying to obtain the real-linear complex representations of $sl(2,C)_R$ from representations of $sl(2,C)$. This is equivalent to obtaining the complex-linear representations of the complexification of $sl(2,C)_R$, $sl_C(2,C)$. So if I find either one I can solve my problem.

Method A: What Wikipedia suggests here is taking the tensor product of 2 known representations of $sl(2,C)$ to obtain a new representation of the complex Lie algebra $sl(2,C)$. This seems to be based on a definition Brian C. Hall also stated in his book. Note I already have the irreducible representations of $su_C(2)$, which means I already have representations of the complex lie algebra $sl(2,C)$. Method A doesn't give me representations of $sl(2,C)_R$ nor $sl_C(2,C)$ (which is what I want), instead only gives me more representations of just $sl(2,C)$. If applying Defintion 4.23 would give me at least the rep. of $sl_C(2,C)$ I could see why is A helpful.

Question 1) Why is A necessary to obtain rep of $sl(2,C)_R$?

If I apply Wikipedia's method I get the following:

Let $(\pi_m,V_m)$ and $(\pi_n,V_n)$ be the irreducible representations (of dimensions $2m+1$ and $2n+1$ respectively) of $su_C(2)$ , and since $su_C(2)\cong sl(2,C)$, then these irreps also describe irreps for $sl(2,C)$. In the other hand, let $(\bar{\pi}_n, \bar{V}_n)$ be the conjugate representation of $(\pi_n,V_n)$ of $sl(2,C)$.

A representation of $sl(2,C)$ over a vector space $V_m \otimes \bar{V_n}$ will be given by $\pi_m \otimes \ \bar{\pi}_n$ defined in the following way:

$\pi_m \otimes \ \bar{\pi}_n (X)=\pi_m(X)\otimes I_{2n +1} + I_{2m+1} \otimes \bar{\pi}_n (X)$ for all $X\in sl(2,C)$

There is no extra procedure to arrive to $sl(2,C)_R$, suddenly $\pi_m \otimes \pi_n$ is assumed to be complex-linear representation of $sl_C(2,C)$, and thus, is equivalent to a real-linear rep. of $sl(2,C)_R.$

Now I can see $\{\sigma_1 /2, \sigma_2 /2, \sigma_3 /2 \}$ form a basis for $sl(2,C)$. Then a basis for $sl(2,C)_R$ can be given by $\{\sigma_1 /2, \sigma_2 /2, \sigma_3 /2, i\sigma_1 /2, i\sigma_2 /2, i\sigma_3 /2 \}$.

If $m=1/2$ and $n=0$ I have the (1/2,0) rep of $sl(2,C)_R$

$\sigma_i /2 \mapsto \pi_{1/2} (\sigma_i /2)\otimes I_{1} = \sigma_i /2$

$i \sigma_i /2 \mapsto \pi_{1/2} (i \sigma_i /2)\otimes I_{1} = i \sigma_i /2$

If $m=0$ and $n=1/2$ I have the (0,1/2) rep of $sl(2,C)_R$

$\sigma_i /2 \mapsto I_{1} \otimes \bar{\pi}_{1/2} (\sigma_i /2) = \bar{\sigma}_i /2$

$i \sigma_i /2 \mapsto I_{1} \otimes \bar{\pi}_{1/2} (i \sigma_i /2) = -i\bar{\sigma}_i /2$

If $m=1/2$ and $n=1/2$ I have the (1/2,1/2) rep of $sl(2,C)_R$

$\sigma_i /2 \mapsto \pi_{1/2} (\sigma_i /2)\otimes I_{2}+ I_{2} \otimes \bar{\pi}_{1/2} (\sigma_i /2) = 1/2 ( \sigma_i \otimes I_2 + I_2 \otimes \bar{\sigma}_i )$

$i\sigma_i /2 \mapsto \pi_{1/2} (i \sigma_i /2)\otimes I_{2}+ I_{2} \otimes \bar{\pi}_{1/2} (i \sigma_i /2) = i/2 ( \sigma_i \otimes I_2 - I_2 \otimes \bar{\sigma}_i )$

Method B: I used this to compare with the results I got with Method A.

The following chain of isomorphisms take place

$sl_C(2,C) \cong sl(2,C) \oplus sl(2,C) \cong su_C(2) \oplus su_C(2) \cong so_C(1,3)$

With this in mind I have obtained the complex-linear representations of $so_C(1,3)$, the complexification of the proper lorentz group. Thus, I have the real-linear representations of $so(1,3).$

I know $so(1,3) \cong sl(2,C)_R$ as Lie algebras, so I expected the representations of $so(1,3)$ be equivalent to the representations of $sl(2,C)_R$, in the same way representations of $su_C(2)$ are equivalent to representations of $sl(2,C)$. However, they aren't.

A basis of $so(1,3)$ that satisfy the same commutation relations the basis of $sl(2,C)_R$ did is given by $\{ J_1, J_2, J_3, K_1, K_2, K_3\}$, where the isomorphism between $so(1,3)$ and $sl(2,C)_R$ takes $J_i \mapsto \sigma_i /2$ and $K_i \mapsto i\sigma_i /2$. But I evaluate the (1/2,0), (0,1/2), and (1/2,1/2) representations of $so(1,3)$ and it doesn't map to the same operator the same representations of $sl(2,C)_R$ did.

Question 2) Why can't I use representations of $so(1,3)$ interchangeably with representations of $sl(2,C)_R$ as we did with rep. of $su_C(2)$ and $sl(2,C)$?

Any help would be highly appreciated.

OBS:

the (1/2,0) rep of $so(1,3)$

$J_i \mapsto \sigma_i /2$

$K_i \mapsto -i \sigma_i /2$

the (0,1/2) rep of $so(1,3)$

$J_i \mapsto \sigma_i /2$

$K_i \mapsto i\sigma_i /2$

the (1/2,1/2) rep of $so(1,3)$

$J_i \mapsto 1/2 ( \sigma_i \otimes I_2 + I_2 \otimes \sigma_i )$

$K_i \mapsto i/2 ( I_2 \otimes \sigma_i -\sigma_i \otimes I_2 )$