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<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>There are lots of things to say, but here\'s one:\n\nIn article <412b269b@news.sentex.net>,\nUrs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n\n>So I am\n>wondering if by reverse engineering this construction from loop space\n>language to algebraic higher-group theory, one can find a previously\n>unnoticed sort of "2-group" or something similar which does not have\n>B = -F but has "quasi-abelian" B and non-abelian A.\n\nA strict 2-group is the same as a crossed module: a group G, a group\nH, an action of G on H and a homomorphism t: H -> G, satisfying some\nequations described in my paper.\n\nTaking Lie algebras we see a strict Lie 2-algebra is the same\nas a differential crossed module: a Lie algebra g, a Lie algebra h,\nan action of g on h and a homomorphism dt: h -> g, satisfying some\nequations.\n\nWhen we think about connections in this context, we see the natural\nequation is not\n\nB = -F\n\n(which makes no sense!) but instead\n\nt(B) = -F\n\nwhere F is a g-valued 2-form and B is an h-valued 2-form.\n\nWe can then consider various special cases.\n\nAt one extreme we can have G trivial and H abelian; then the\nabove equation is vacuous. This is what happens in 2-form\nelectromagnetism.\n\nAt another extreme we can have G = H and t the identity; then\nyou get B = -F. That\'s the case you seem to like best.\n\nBut there are lots of intermediate cases. Maybe you want some\nconcrete examples? I can manufacture examples, but not very\ninteresting ones if G and H are required to be *compact* Lie groups,\nbecause the Lie algebra of these is always semisimple + abelian,\nand the options for homomorphisms dt are severely limited.\n\n(As usual I will cc this to you.)\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>There are lots of things to say, but here's one:
In article <412b269b@news.sentex.net>,
Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
>So I am
>wondering if by reverse engineering this construction from loop space
>language to algebraic higher-group theory, one can find a previously
>unnoticed sort of "2-group" or something similar which does not have
[itex]>B = -F[/itex] but has "quasi-abelian" B and non-abelian A.
A strict 2-group is the same as a crossed module: a group G, a group
H, an action of G on H and a homomorphism t: [itex]H -> G,[/itex] satisfying some
equations described in my paper.
Taking Lie algebras we see a strict Lie 2-algebra is the same
as a differential crossed module: a Lie algebra g, a Lie algebra h,
an action of g on h and a homomorphism dt: [itex]h -> g,[/itex] satisfying some
equations.
When we think about connections in this context, we see the natural
equation is not
[tex]B = -F[/tex]
(which makes no sense!) but instead
t(B) [itex]= -F[/itex]
where F is a g-valued 2-form and B is an h-valued 2-form.
We can then consider various special cases.
At one extreme we can have G trivial and H abelian; then the
above equation is vacuous. This is what happens in 2-form
electromagnetism.
At another extreme we can have [itex]G = H[/itex] and t the identity; then
you get [itex]B = -F[/itex]. That's the case you seem to like best.
But there are lots of intermediate cases. Maybe you want some
concrete examples? I can manufacture examples, but not very
interesting ones if G and H are required to be *compact* Lie groups,
because the Lie algebra of these is always semisimple + abelian,
and the options for homomorphisms dt are severely limited.
(As usual I will cc this to you.)
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