Extending a linear representation by an anti-linear operator

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Main Question or Discussion Point

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 isnt 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.
 

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