Wikipedia:Reference desk/Archives/Mathematics/2014 July 10

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July 10

Lie algebras of U(p, q)

Hi!

Which, if any, is correct?

${\mathfrak {u}}(p,q)=\left\{\left.\left({\begin{matrix}X_{p\times p}&Z_{p\times q}\\-{\overline {Z}}^{\rm {T}}&Y_{q\times q}\end{matrix}}\right)\right|{\overline {X}}^{\rm {T}}=-X,\quad {\overline {Y}}^{\rm {T}}=-Y\right\}.$

${\mathfrak {u}}(p,q)=\left\{\left.\left({\begin{matrix}X_{p\times p}&Z_{p\times q}\\+{\overline {Z}}^{\rm {T}}&Y_{q\times q}\end{matrix}}\right)\right|{\overline {X}}^{\rm {T}}=-X,\quad {\overline {Y}}^{\rm {T}}=-Y\right\}.$

I have one opinion and my reference another about the +/- sign in the lower left. YohanN7 (talk) 21:29, 10 July 2014 (UTC)

We only have a section on indefinite unitary groups and less on the Lie algebras, but if you know the Lie group it's generally easy enough to work out corresponding tangents at I. Take

$\left({\begin{matrix}I+tX&tZ\\tW&I+tY\end{matrix}}\right)$

to be in the unitary group U(p, q). It must preserve the form giving

$(I+tX){\overline {(I+tX)}}^{\intercal }-(tZ){\overline {(tZ)}}^{\intercal }=I\,$

$(I+tX){\overline {(tW)}}^{\intercal }-(tZ){\overline {(I+tY)}}^{\intercal }=0\,$

$(tW){\overline {(I+tX)}}^{\intercal }-(I+tY){\overline {(tZ)}}^{\intercal }=0\,$

$(tW){\overline {(tW)}}^{\intercal }-(I+tY){\overline {(I+tY)}}^{\intercal }=-I.$

Now take derivatives and evaluate at t=0 to get

$X+{\overline {X}}^{\intercal }=0\,$

${\overline {W}}^{\intercal }-Z=0\,$

$W-{\overline {Z}}^{\intercal }=0\,$

$-Y-{\overline {Y}}^{\intercal }=0\,$

in other words, according to my calculations, the plus sign is correct.

{Btw, I thought U(3, 1) was a big deal in physics, so why don't we have more material on it?)--RDBury (talk) 15:16, 11 July 2014 (UTC)

Thank you, your calculation confirms mine, done in a different way. We will in a day or two have at least something on the indefinite unitary groups and other classical groups. (This is why I asked in the first place.) I am making a complete rewrite of Classical group. Today's article manages to miss the majority of the classical groups (while it does contain stuff about fields not of characteristic 0, hardly in the spirit of classical groups). You can find my version from my user page. (My draft has a talk page if anyone wants to comment.) I expect to be done in a day or two. YohanN7 (talk) 16:26, 11 July 2014 (UTC)

Actually, the new version is in place now. YohanN7 (talk) 18:37, 11 July 2014 (UTC)