- #1

- 41

- 1

I recently got confused about Lie group products.

Say, I have a group [itex]U(1)\times U(1)'[/itex]. Is this group reducible into two [itex]U(1)[/itex]'s, i.e. possible to resepent with a matrix [itex]\rho(U(1)\times U(1)')=\rho_{1}(U(1))\oplus\rho_{1}(U(1)')=e^{i\theta_{1}}\oplus e^{i\theta_{2}}=\begin{pmatrix}e^{i\theta_{1}} & 0 \\ 0 & e^{i\theta_{2}}\end{pmatrix}[/itex]? Can I say it's reducible, right? Because the way I see it, if the transformation is applied to a 2-dimensional vector, then the first (second) element is transformed by the first (second) [itex]U(1)[/itex] ([itex]U(1)'[/itex]), thus leaving us two invariant 1-dimensional subspaces under the group actions.

Is it always possible to represent a group product as the direct sum of individual group representations? Or is it just an Abelian case? (IMHO, it seems so because the transformation [itex]SU(2)\times U(1)[/itex] on leptons isn't a [itex]3\times3[/itex] block-diagonal matrix (as one would expect, because fundamental rep. dimensions are 2+1 = 3) but a [itex]2\times 2[/itex] matrix).

Thanks a lot

edit: bonus question -- is [itex]2\times2[/itex] rep. of [itex]U(1)[/itex], [itex]\begin{pmatrix}e^{i\theta} & 0 \\ 0 & e^{i\theta}\end{pmatrix}[/itex] a reducible or irreducible representation?

Say, I have a group [itex]U(1)\times U(1)'[/itex]. Is this group reducible into two [itex]U(1)[/itex]'s, i.e. possible to resepent with a matrix [itex]\rho(U(1)\times U(1)')=\rho_{1}(U(1))\oplus\rho_{1}(U(1)')=e^{i\theta_{1}}\oplus e^{i\theta_{2}}=\begin{pmatrix}e^{i\theta_{1}} & 0 \\ 0 & e^{i\theta_{2}}\end{pmatrix}[/itex]? Can I say it's reducible, right? Because the way I see it, if the transformation is applied to a 2-dimensional vector, then the first (second) element is transformed by the first (second) [itex]U(1)[/itex] ([itex]U(1)'[/itex]), thus leaving us two invariant 1-dimensional subspaces under the group actions.

Is it always possible to represent a group product as the direct sum of individual group representations? Or is it just an Abelian case? (IMHO, it seems so because the transformation [itex]SU(2)\times U(1)[/itex] on leptons isn't a [itex]3\times3[/itex] block-diagonal matrix (as one would expect, because fundamental rep. dimensions are 2+1 = 3) but a [itex]2\times 2[/itex] matrix).

Thanks a lot

edit: bonus question -- is [itex]2\times2[/itex] rep. of [itex]U(1)[/itex], [itex]\begin{pmatrix}e^{i\theta} & 0 \\ 0 & e^{i\theta}\end{pmatrix}[/itex] a reducible or irreducible representation?

Last edited: