theos ek mechanes
Aug2-04, 06:11 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>I am generally a group theory idiot; so, when\nanswering keep that in mind. My question deals\nwith ISL(2,C) and more to the point isl(2,C).\n\nISL(2,C) can be viewed as the semi-direct product\nof SL(2,C) and H(2). H(2) being the set of all 2x2\nHermitian matrices viewed as a abelian group with\nthe group product being matrix addition. The\nsemi-direct product comes as a result of the\nnatural action of SL(2,C) on H(2). For example,\nif L^A_B is an element of SL(2,C) and P^{AA\'}\nis an element of H(2), then the natural action\nof SL(2,C) on H(2) is given by\n\nL^A_B {\\overline L}^{A\'}_{B\'} P^{BB\'}\n\nHence, the group product in ISL(2,C) is given by\n\n(L^A_B, P^{AA\'}) x (M^A_B, N^{AA\'}) =\n\n(L^A_B M^B_C, P^{AA\'} + L^A_B {\\overline L}^{A\'}_{B\'} N^{BB\'})\n\nOk. So far, so good...\n\nNow going to isl(2,C). Over the reals isl(2,C)\nis of dimension 10 with 6 generators, each of\nthe form l^A_B, coming from sl(2,C) and 4, each\nof the form p^{AA\'}, coming from from H(2).\n\nMy question is what invariant, non-degenerate\nmetrics exist on the Lie Algebra isl(2,C). Or\nI guess one could equivalently ask for what\ninvariant, non-degenerate Killing forms exist\non isl(2,C)...\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>I am generally a group theory idiot; so, when
answering keep that in mind. My question deals
with ISL(2,C) and more to the point isl(2,C).
ISL(2,C) can be viewed as the semi-direct product
of SL(2,C) and H(2). H(2) being the set of all 2x2
Hermitian matrices viewed as a abelian group with
the group product being matrix addition. The
semi-direct product comes as a result of the
natural action of SL(2,C) on H(2). For example,
if L^{A_B} is an element of SL(2,C) and P^{AA'}
is an element of H(2), then the natural action
of SL(2,C) on H(2) is given by
L^{A_B} {\overline L}^{A'}_{B'} P^{BB'}
Hence, the group product in ISL(2,C) is given by
(L^{A_B}, P^{AA'}) x (M^{A_B}, N^{AA'}) =(L^{A_B} M^{B_C}, P^{AA'} + L^{A_B} {\overline L}^{A'}_{B'} N^{BB'})
Ok. So far, so good...
Now going to isl(2,C). Over the reals isl(2,C)
is of dimension 10 with 6 generators, each of
the form l^{A_B}, coming from sl(2,C) and 4, each
of the form p^{AA'}, coming from from H(2).
My question is what invariant, non-degenerate
metrics exist on the Lie Algebra isl(2,C). Or
I guess one could equivalently ask for what
invariant, non-degenerate Killing forms exist
on isl(2,C)...
answering keep that in mind. My question deals
with ISL(2,C) and more to the point isl(2,C).
ISL(2,C) can be viewed as the semi-direct product
of SL(2,C) and H(2). H(2) being the set of all 2x2
Hermitian matrices viewed as a abelian group with
the group product being matrix addition. The
semi-direct product comes as a result of the
natural action of SL(2,C) on H(2). For example,
if L^{A_B} is an element of SL(2,C) and P^{AA'}
is an element of H(2), then the natural action
of SL(2,C) on H(2) is given by
L^{A_B} {\overline L}^{A'}_{B'} P^{BB'}
Hence, the group product in ISL(2,C) is given by
(L^{A_B}, P^{AA'}) x (M^{A_B}, N^{AA'}) =(L^{A_B} M^{B_C}, P^{AA'} + L^{A_B} {\overline L}^{A'}_{B'} N^{BB'})
Ok. So far, so good...
Now going to isl(2,C). Over the reals isl(2,C)
is of dimension 10 with 6 generators, each of
the form l^{A_B}, coming from sl(2,C) and 4, each
of the form p^{AA'}, coming from from H(2).
My question is what invariant, non-degenerate
metrics exist on the Lie Algebra isl(2,C). Or
I guess one could equivalently ask for what
invariant, non-degenerate Killing forms exist
on isl(2,C)...