A How can the LRS model for leptons incorporate the Standard Model group?

[Shen712]

This is a homework problem in a course in particle physics at Cornell University.
Assume the Left Right Symmetric (LRS) model for leptons. The gauge group is G_LR = SU(2)_L×SU(2)_R×U(1)_X. The Standard Model group SU(2)_L×U(1)_Y has to be included in the LRS group. Namely, U(1)_Y ⊂ SU(2)_R×U(1)_X. Find the linear combination of the LRS generators which gives the Standard Model generator Y.
The answer is: Y = T₃^R + X/2
How to get this result?

 

[Orodruin]

To give away the answer directly would violate Physics Forums rules. What are your own thoughts and what have you been able to conclude so far?

 

[Shen712]

To give away the answer directly would violate Physics Forums rules. What are your own thoughts and what have you been able to conclude so far?

My thoughts are: Since the SM generator Y is proportional to identity, the required linear combination of the LRS generators must also be proportional to identity. Considering that the generators of SU(2)_R are T_a^R = τ_a/2, where τ_a are the Pauli matrices, only the combination of T₃^R and X is possible to be proportional to identity. Thus, Y = T₃^R + kX, where k is some constant. Since T_a^R = τ_a/2, if we take k = 1/2, we will get Y = τ_a/2 + X/2 = diag(1+X, -1+X)/2. This is a diagonalized matrix, but not proportional to identity. I am a little puzzled.
By the way, I am not a student at Cornell University, and this course is an old one in 2008. I just downloaded the course materials from the website and study them by myself. In this case, do the Physics Forum rules allow to give away the answer?
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[jtbell]

In this case, do the Physics Forum rules allow to give away the answer?

No. We believe that regardless of whether you're doing an exercise as part of a course for credit, or for independent self-study, you're best served by figuring out the answer for yourself, with some help from hints and/or corrections as appropriate.

 

[ChrisVer]

I suppose that since you are looking at models beyond the standard model, you have already looked at the standard model itself... haven't you?
If yes, the problem is pretty much the same as the EM charge in the Electroweak theory: SU_L(2) \times U_Y(1) \rightarrow U_{Q}(1).