Extending a linear representation by an anti-linear operator

[hideelo]

When we consider representation of the O(1,3) we divide the representation up into the connected part of SO(1,3) which we then extend by adding in the discrete transformations P and T. However, while (as far as I know) we always demand that the SO(1,3) and P be represented as linear operators, we demand that T be anti-linear.

If I start with some representation of SO(1,3) and I extended it by including T. Since its anti linear and anti unitary it can be written as T = CU where U is unitary and C is the complex conjugation operator. Suppose U were already in SO(1,3), then while I say I am adding T really the only new object I am adding is C. So suppose I have any representation and I extend by C, what is this new object?

I am just looking for the mathematical framework in which to approach this. Once I add T it isn't a linear representation anymore, I see that we have some operators which are homomorphic to O(1,3) but this isn't a representation as I recognize it. So what is this?

While I am framing all this in the context of SO(1,3), it's only because that is how it arose for me. Really, I'm interested in the general question. Suppose I had some linear representation of some group which I then extend by the complex conjugation operator, thereby allowing anti-linear operators. Is this a representation of some bigger group?Any references, sources, etc would be really appreciated.

 

[fresh_42]

I usually start with the definitions. A representation of $O(1,3)$ is a group homomorphism into $\operatorname{GL}(V)$. As the orthogonal group is already a matrix group, we get one representation for free.

You could also have a look into:
https://www.amazon.com/dp/0387901086/?tag=pfamazon01-20