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Orbis T.
Apr22-04, 03:06 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>Writing the antisymmetric (f_{jkl}) and symmetric\n(d_{jkl}) structure constants of the Lie algebra of\nSU(N) (determined by the generators F_{m}) as follows:\n\nf_{jkl} = - (i / 4) Tr([F_{j},F_{k}]_{-} F_{l})\n\nand\n\nd_{jkl} = (1 / 4) Tr([F_{j},F_{k}]_{+} F_{l})\n\nrespectively, one can then obtain the following\nidentities:\n\na) f_{jkl} d_{lmn} + f_{nkl} d_{lmj} + f_{mkl}d_{jln}\n= 0\n\nb) f_{jkl} f_{lmn} + f_{mkl} f_{lnj} + f_{jml}f_{lnk}\n= 0\n\nThe identity a) contains both symmetric and\nantisymmetric structure constants while b) contains\nonly antisymmetric constants.\n\nIs there some identity of the kind of b) (or similar)\nbut involving only the symmetric structure constants?\nCould you recommend me a good book where I could learn\na bit about general properties of the Lie algebra of\nSU(N) (I have been reading some books of Group Theory\nfor physicists but they are usually restricted to the\ncase N \\leq 5).\n\nbest,\n\nOrbis\n\n\n\n\n\n_________________ _________________\nDo you Yahoo!?\nYahoo! Photos: High-quality 4x6 digital prints for 25¢\nhttp://photos.yahoo.com/ph/print_splash\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">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>Writing the antisymmetric (f_{jkl}) and symmetric
(d_{jkl}) structure constants of the Lie algebra of
SU(N) (determined by the generators F_{m}) as follows:

f_{jkl} = - (i / 4) Tr([F_{j},F_{k}]_{-} F_{l})

and

d_{jkl} = (1 / 4) Tr([F_{j},F_{k}]_{+} F_{l})

respectively, one can then obtain the following
identities:

a) f_{jkl} d_{lmn} + f_{nkl} d_{lmj} + f_{mkl}d_{jln}
=

b) f_{jkl} f_{lmn} + f_{mkl} f_{lnj} + f_{jml}f_{lnk}
=

The identity a) contains both symmetric and
antisymmetric structure constants while b) contains
only antisymmetric constants.

Is there some identity of the kind of b) (or similar)
but involving only the symmetric structure constants?
Could you recommend me a good book where I could learn
a bit about general properties of the Lie algebra of
SU(N) (I have been reading some books of Group Theory
for physicists but they are usually restricted to the
case N \leq 5).

best,

Orbis





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Alfred Einstead
Apr29-04, 12:47 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>\n\n\n\n"Orbis T." &lt;orbis_t@yahoo.com&gt; wrote:\n&gt; Is there some identity of the kind of b) (or similar)\n&gt; but involving only the symmetric structure constants?\n&gt; Could you recommend me a good book where I could learn\n&gt; a bit about general properties of the Lie algebra of\n&gt; SU(N) (I have been reading some books of Group Theory\n&gt; for physicists but they are usually restricted to the\n&gt; case N \\leq 5).\n\nIt\'s easier to work with GL(N,C) first. The generators are\njust the unit matrices e^i_j with the product rule:\ne^i_j e^k_l = delta^k_j e^i_l; 0 &lt;= i,j,l,k &lt; N\nand commutator\n[e^i_j, e^k_l] = delta^k_j e^i_l - delta^i_l e^k_j,\nwhere delta is the Kroenecker delta.\n\nThis means that\n[e^i_j, e^k_l] = 0 unless i = l or j = k,\nand the following examples illustrate all the essential cases\nwhere i = l or j = k:\n[e^1_1, e^1_2] = e^1_2\n[e^1_1, e^2_1] = -e^2_1\n[e^1_1, e^1_1] = 0\n[e^1_2, e^1_1] = -e^1_2\n[e^1_2, e^2_2] = e^1_2\n[e^1_2, e^2_1] = e^1_1 - e^2_2\n[e^1_2, e^2_3] = e^1_3\n[e^1_2, e^3_1] = -e^3_1\n\nFor SU(N), the generators are\nA^k_l = e^k_l + e^l_k; 0 &lt;= k &lt; l &lt; N\nB^k_l = (e^k_l - e^l_k)/i; 0 &lt;= k &lt; l &lt; N\nand\nC^k = (e^0_0 + ... + e^{k-1}_{k-1} - k e^k_k)/sqrt(k(k+1)/2)\nfor 0 &lt; k &lt; N.\n\nUsing the formula above for the unit matrix commutators you should\nbe able to figure out for yourself the commutators for the A\'s, B\'s\nand C\'s; out of which the structure constants come.\n\nFor instance\n[A^1_2, A^2_3] = [e^1_2 + e^2_1, e^2_3 + e^3_2]\n= e^1_3 - e^3_1\n= i B^1_3.\n[B^1_2, B^2_3] = - [e^1_2 - e^2_1, e^2_3 - e^3_2]\n= -e^1_3 + e^3_1\n= -i B^1_3\n[A^1_2, B^2_3] = -i [e^1_2 + e^2_1, e^2_3 - e^3_2]\n= -i e^1_3 - i e^3_1\n= -i A^1_3\n[B^1_2, A^2_3] = -i [e^1_2 - e^2_1, e^2_3 + e^3_2]\n= -i e^1_3 - i e^3_1\n= -i A^1_3.\n\nThe most interesting case is for SU(3), where the generators\nare\nA^1_2, B^1_2, A^1_3, B^1_3, A^2_3, B^1_3, C^1 and C^2.\nIt\'s actually more revealing to write the A\'s and B\'s in terms of\nthe corresponding unit matrices\n(A^1_2 + i B^1_2)/2 = e^1_2 = R\n(A^1_3 + i B^1_3)/2 = e^1_3 = O\n(A^2_3 + i B^2_3)/2 = e^2_3 = Y\n(A^1_2 - i B^1_2)/2 = e^2_1 = G\n(A^1_3 - i B^1_3)/2 = e^3_1 = B\n(A^2_3 - i B^2_3)/2 = e^3_2 = P\nand the C matrices\nC^1 = (e^1_1 - e^2_2)\nC^2 = (e^1_1 + e^2_2 - 2 e^3_3)/sqrt(3)\nas the redundant set\nU = C^1 = e^1_1 - e^2_2\nV = -1/2 C^1 + sqrt(3)/2 C^2 = e^2_2 - e^3_3\nW = -1/2 C^1 - sqrt(3)/2 C^2 = e^3_3 - e^1_1.\nThen you get:\n[R,O] = [O,Y] = [Y,G] = [G,B] = [B,P] = [P,R] = 0,\n[R,Y] = O, [Y,B] = G, [B,R] = P\n[P,G] = B, [G,O] = Y, [O,P] = R\n[R,G] = U, [Y,P] = V, [B,O] = W\n[U,V] = [V,W] = [W,U] = 0; with U + V + W = 0\nand\n[x,y] = k_{xy} y; for x=U,V,W; y=R,O,Y,G,B,P\nwith k given by the following table\nR O Y G B P\nU 2 1 -1 -2 -1 1\nV -1 1 2 1 -1 -2\nW -1 -2 -1 1 2 1.\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">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Orbis T." <orbis_t@yahoo.com> wrote:
> Is there some identity of the kind of b) (or similar)
> but involving only the symmetric structure constants?
> Could you recommend me a good book where I could learn
> a bit about general properties of the Lie algebra of
> SU(N) (I have been reading some books of Group Theory
> for physicists but they are usually restricted to the
> case N \leq 5).

It's easier to work with GL(N,C) first. The generators are
just the unit matrices e^{i_j} with the product rule:
e^{i_j} e^{k_l} = \delta^k_j e^{i_l}; <= i,j,l,k < N
and commutator
[e^{i_j}, e^{k_l}] = \delta^k_j e^{i_l} - \delta^i_l e^{k_j},
where \delta is the Kroenecker \delta.

This means that
[e^{i_j}, e^{k_l}] = unless i = l or j = k,
and the following examples illustrate all the essential cases
where i = l or j = k:[e^{1_1}, e^{1_2}] = e^{1_2}[e^{1_1}, e^{2_1}] = -e^{2_1}[e^{1_1}, e^{1_1}] =[e^{1_2}, e^{1_1}] = -e^{1_2}[e^{1_2}, e^{2_2}] = e^{1_2}[e^{1_2}, e^{2_1}] = e^{1_1} - e^{2_2}[e^{1_2}, e^{2_3}] = e^{1_3}[e^{1_2}, e^{3_1}] = -e^{3_1}

For SU(N), the generators are
A^{k_l} = e^{k_l} + e^{l_k};<= k < l < NB^{k_l} = (e^{k_l} - e^{l_k})/i;<= k < l < N
and
C^k = (e^{0_0} + ... + e^{k-1}_{k-1} - k e^{k_k})/\sqrt(k(k+1)/2)
for < k < N.

Using the formula above for the unit matrix commutators you should
be able to figure out for yourself the commutators for the A's, B's
and C's; out of which the structure constants come.

For instance
[A^{1_2}, A^{2_3}] = [e^{1_2} + e^{2_1}, e^{2_3} + e^{3_2}]= e^{1_3} - e^{3_1}= i B^{1_3}.[B^{1_2}, B^{2_3}] = - [e^{1_2} - e^{2_1}, e^{2_3} - e^{3_2}]= -e^{1_3} + e^{3_1}= -i B^{1_3}[A^{1_2}, B^{2_3}] = -i [e^{1_2} + e^{2_1}, e^{2_3} - e^{3_2}]= -i e^{1_3} - i e^{3_1}= -i A^{1_3}[B^{1_2}, A^{2_3}] = -i [e^{1_2} - e^{2_1}, e^{2_3} + e^{3_2}]= -i e^{1_3} - i e^{3_1}= -i A^{1_3}.

The most interesting case is for SU(3), where the generators
are
A^{1_2}, B^{1_2}, A^{1_3}, B^{1_3}, A^{2_3}, B^{1_3}, C^1 and C^2.
It's actually more revealing to write the A's and B's in terms of
the corresponding unit matrices
(A^{1_2} + i B^{1_2})/2 = e^{1_2} = R(A^{1_3} + i B^{1_3})/2 = e^{1_3} = O(A^{2_3} + i B^{2_3})/2 = e^{2_3} = Y(A^{1_2} - i B^{1_2})/2 = e^{2_1} = G(A^{1_3} - i B^{1_3})/2 = e^{3_1} = B(A^{2_3} - i B^{2_3})/2 = e^{3_2} = P
and the C matrices
C^1 = (e^{1_1} - e^{2_2})C^2 = (e^{1_1} + e^{2_2} - 2 e^{3_3})/\sqrt(3)
as the redundant set
U = C^1 = e^{1_1} - e^{2_2}V = -1/2 C^1 + \sqrt(3)/2 C^2 = e^{2_2} - e^{3_3}W = -1/2 C^1 - \sqrt(3)/2 C^2 = e^{3_3} - e^{1_1}.
Then you get:
[R,O] = [O,Y] = [Y,G] = [G,B] = [B,P] = [P,R] = 0,
[R,Y] = O, [Y,B] = G, [B,R] = P
[P,G] = B, [G,O] = Y, [O,P] = R
[R,G] = U, [Y,P] = V, [B,O] = W
[U,V] = [V,W] = [W,U] = 0; with U + V + W =
and
[x,y] = k_{xy} y; for x=U,V,W; y=R,O,Y,G,B,P
with k given by the following table
R O Y G B P
U 2 1 -1 -2 -1 1
V -1 1 2 1 -1 -2W -1 -2 -1 1 2 1.