<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>\n\nI know what a Lie superalgebra is and I also know that the tangent\nspace of a Lie group at its identity is its Lie group. What what is a\nLie supergroup? I can\'t find this answer anywhere!\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 know what a Lie superalgebra is and I also know that the tangent
space of a Lie group at its identity is its Lie group. What what is a
Lie supergroup? I can't find this answer anywhere!
Michael Murray
Jul23-04, 08:30 AM
<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>\nIn article <3cb3ea56.0407221024.73378579@posting.google.com>,\nvery_cryptic@hotmail.com (Very cryptic) wrote:\n\n> I know what a Lie superalgebra is and I also know that the tangent\n> space of a Lie group at its identity is its Lie group. What what is a\n> Lie supergroup? I can\'t find this answer anywhere!\n\nThere is at least one book I can find with Google which is\n\nhttp://www2.springeronline.com/sgw/cda/frontpage/0,,5-10043-22-33664019-0\n,00.html\n\nYou have to first decided what a supermanifold is. I haven\'t thought\nabout this for quite a long time. Last time I looked at it there were\ntwo approaches\n\n(1) Replace the R^n in the definition of manifold by a Grassman algebra\nor a graded Banach algebra\n\n(2) Do something to the structure sheaf of a manifold to expand\nit to include non-commuting functions.\n\nMichael\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>In article <3cb3ea56.0407221024.73378579@posting.google.com>,
very_cryptic@hotmail.com (Very cryptic) wrote:
> I know what a Lie superalgebra is and I also know that the tangent
> space of a Lie group at its identity is its Lie group. What what is a
> Lie supergroup? I can't find this answer anywhere!
There is at least one book I can find with Google which is
You have to first decided what a supermanifold is. I haven't thought
about this for quite a long time. Last time I looked at it there were
two approaches
(1) Replace the R^n in the definition of manifold by a Grassman algebra
or a graded Banach algebra
(2) Do something to the structure sheaf of a manifold to expand
it to include non-commuting functions.
Michael
Very cryptic
Jul25-04, 09:16 AM
<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>\n\n\n\nI\'d like to clarify a couple of things. I know what a supermanifold is\n(with the coordinates taking on values in a Grassman algebra) and I do\nknow one of the definitions of a Lie supergroup is a differentiable\nsupermanifold with a smooth (analytic) group structure. However, the\nrepresentations of such a supergroup act upon vector spaces over a\nGrassman algebra. When I think of a Lie superalgebra, I think of a\nreal/complex Z_2 graded algebra with its representations acting upon\nZ_2 graded real/complex vector spaces. I guess I should have been more\nprecise in my earlier question. What is the analog of the Lie group\nfor a real/complex Z_2 graded Lie superalgebra (not a Lie superalgebra\nover a Grassman algebra)? The reason I\'m asking this is because I\'d\nlike the representations of such a Lie supergroup to act upon Z_2\ngraded real/complex vector spaces instead of a vector superspace.\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'd like to clarify a couple of things. I know what a supermanifold is
(with the coordinates taking on values in a Grassman algebra) and I do
know one of the definitions of a Lie supergroup is a differentiable
supermanifold with a smooth (analytic) group structure. However, the
representations of such a supergroup act upon vector spaces over a
Grassman algebra. When I think of a Lie superalgebra, I think of a
real/complex Z_2 graded algebra with its representations acting upon
Z_2 graded real/complex vector spaces. I guess I should have been more
precise in my earlier question. What is the analog of the Lie group
for a real/complex Z_2 graded Lie superalgebra (not a Lie superalgebra
over a Grassman algebra)? The reason I'm asking this is because I'd
like the representations of such a Lie supergroup to act upon Z_2
graded real/complex vector spaces instead of a vector superspace.