I recently got confused about Lie group products.(adsbygoogle = window.adsbygoogle || []).push({});

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?

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# I About Lie group product ([itex]U(1)\times U(1)[/itex] ex.)

Have something to add?

Draft saved
Draft deleted

Loading...

Similar Threads - group product itex | Date |
---|---|

Quotient of group by a semidirect product of subgroups | Feb 18, 2016 |

Two quotient groups implying Cartesian product? | Apr 5, 2015 |

Direct Product of Cyclic Groups | Nov 6, 2014 |

Product Groups and their dimensions | Jul 23, 2012 |

Direct Product of Groups | Jun 17, 2012 |

**Physics Forums - The Fusion of Science and Community**