Henriques: "A Lie algebra for the String group"

Feb1-05, 08:20 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>Urs Schreiber wrote:\n\n> Cook up any action principle which involves a non-associative operation at\n> one point or another. Quantize this action. The resulting quantum theory\n> will not involve non-associative quantum operators. Similarly, the\n> quantization of a nonlinear field theory will not involve non-linear quantum\n> operators.\n\nI\'m not sure that I can cook up any action principle which involves a\nnon-associative operation. Something with octonions, perhaps.\nQuantization of such a structure seems to be very subtle. It is definitely\nnot a textbook case like phi^4 or other nonlinear field theories.\n\nBesides, how can classical non-associativity go away after quantization.\nLet\'s go the other way. You have some associative product (a star\nproduct probably), and let hbar -> 0. In this limit a non-associativity\narises. Why?\n\n>\n> One example are certain flavors of string field theory, which do involve\n> non-associative operations in their action.\n\nHm, hm. That ain\'t impressing me much, as Shania Twain would have said.\n\n>\n> Then, it is remarkable that Henriques\' construction does not really involve\n> any non-associativity after all. The bracket\n>\n> [(X,a), (Y,b)] := ([X,Y], 0)\n>\n> satisfies the Jacobi identity on the nose.\n>\n> The Jacobiator in this example degenerates to an automorphism 2-morphism on\n> the on 1-morphism [[(X,a),(Y,b)],(Z,c)].\n\nThe Jacobi identity holds of course for the bilinear bracket, but how\ncan then [(X,a),(Y,b),(Z,c)] != 0 ? One can choose the structure\nconstants, and thus <[X,Y],Z>, to be totally skew.\n\nIf you antisymmetrize [[(X,a),(Y,b)],(Z,c)], you get zero, no?\n\nIs not [(X,a),(Y,b),(Z,c)] totally skew?\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>Urs Schreiber wrote:

> Cook up any action principle which involves a non-associative operation at
> one point or another. Quantize this action. The resulting quantum theory
> will not involve non-associative quantum operators. Similarly, the
> quantization of a nonlinear field theory will not involve non-linear quantum
> operators.

I'm not sure that I can cook up any action principle which involves a
non-associative operation. Something with octonions, perhaps.
Quantization of such a structure seems to be very subtle. It is definitely
not a textbook case like $LaTeX Code: \\phi^4$ or other nonlinear field theories.

Besides, how can classical non-associativity go away after quantization.
Let's go the other way. You have some associative product (a star
product probably), and let $LaTeX Code: \\hbar ->$ . In this limit a non-associativity
arises. Why?

>
> One example are certain flavors of string field theory, which do involve
> non-associative operations in their action.

Hm, hm. That ain't impressing me much, as Shania Twain would have said.

>
> Then, it is remarkable that Henriques' construction does not really involve
> any non-associativity after all. The bracket
>
> [(X,a), (Y,b)]

>
> satisfies the Jacobi identity on the nose.
>
> The Jacobiator in this example degenerates to an automorphism 2-morphism on
> the on 1-morphism [[(X,a),(Y,b)],(Z,c)].

The Jacobi identity holds of course for the bilinear bracket, but how
can then [(X,a),(Y,b),(Z,c)] != ? One can choose the structure
constants, and thus <[X,Y],Z>, to be totally skew.

If you antisymmetrize [[(X,a),(Y,b)],(Z,c)], you get zero, no?

Is not [(X,a),(Y,b),(Z,c)] totally skew?

Feb1-05, 08:58 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>On Tue, 1 Feb 2005, Thomas Larsson wrote:\n\n> Urs Schreiber wrote:\n>\n> > Cook up any action principle which involves a non-associative operation at\n> > one point or another. Quantize this action. The resulting quantum theory\n> > will not involve non-associative quantum operators. Similarly, the\n> > quantization of a nonlinear field theory will not involve non-linear quantum\n> > operators.\n>\n> I\'m not sure that I can cook up any action principle which involves a\n> non-associative operation.\n\n\nLet\n\n(.,.) : R^2 -> R\n(x,y) |-> x y^2\n\nbe a non-associative product on the reals.\nLet\n\n\nx : R -> R\nt |-> x(t)\n\nbe the trajectory of some non-relativistic particle, being\nan element of some suitable function space F and consider\nthe action\n\n\nS : F -> R\nx |-> int ( (dx(t)/dt)^2 - (x(t),x(t)) ) dt .\n\n\nIt is as simple as that. An action is a map from the space of\nconfigurations to the real numbers. The nature of the theory\nobtained by taking the path integral using that action does\nnot depend on the choice of algorith by which you may want\nto compute the number associated to a given configuration,\ni.e. the algorithm by which you want to compute the value\nof S on a given configuration x.\n\nWe had a similar discussion some time ago on spr. There you\nwere concerned about noncommutative field theories. In that case,\ntoo, the noncommutativity of any product that appears in\nthe formula for computing the action does not affect the\nproperties of the product of the operators on the Hilbert\nspace of the quantum theory obtained from that action.\n\n\n\n> > One example are certain flavors of string field theory, which do involve\n> > non-associative operations in their action.\n>\n> Hm, hm. That ain\'t impressing me much,\n\n\n\nIt is an example that you were asking for.\n\n\n\n> > Then, it is remarkable that Henriques\' construction does not really involve\n> > any non-associativity after all. The bracket\n> >\n> > [(X,a), (Y,b)] := ([X,Y], 0)\n> >\n> > satisfies the Jacobi identity on the nose.\n> >\n> > The Jacobiator in this example degenerates to an automorphism 2-morphism on\n> > the on 1-morphism [[(X,a),(Y,b)],(Z,c)].\n>\n> The Jacobi identity holds of course for the bilinear bracket, but how\n> can then [(X,a),(Y,b),(Z,c)] != 0 ?\n\n\n\nBecause the trinary bracket is not equal to the Jacobi operation on the\nbinary bracket in a 2-term L_oo algebra. Instead, there is also a\nunary operation d(.) and one of the relations to be satisfied by a\n2-term L_oo algebra (or \'L_2 algebra\') is\n\nd[x,y,z] = [x,[y,z]] + [[x,z],y] - [[x,y],z] .\n\nwhere x stands for (x,0) etc.\n\nAnother relation is\n\n[x,y,d(h)] = [x,[y,d(h)]] + [[x,d(h)],y] - [[x,y],d(h)] .\n\nwhere h stands for (0,h).\n\nComparing these two relations with the brackets that we are given\none concludes that d must act trivially by sending everything to 0,\nbecause the right hand side of both these relations vanishes in\nthe present case. This is why I wrote in a previous email that\n\n> where, BTW, I assume that Henriques means to set d(a) = 0\n\nIndeed, Henriques didn\'t bother spelling this out because he is\nconsidering a special example of the algebras discussed\nin example 55 of math.QA/0307263.\n\nYou can find the general definition of an L_2 algebra on\np. 28 of that paper.\n\nThis has an intriguing relation to loop space calculus,\nas explained in\n\nGetzler & Jones, Ill. J. Math. 34,2 (1990), p. 256\n\nand indeed in Hofman\'s paper, the central argument of which\nI review in http://www-stud.uni-essen.de/~sb0264/2conn2.pdf .\n\n\nThe difference between the above L_2 algebra and an ordinary\nLie algebra is rather small. Still, \'exponentiating\' it\ngives something interesting, as discussed in section 8.5 of\nmath.QA/0307200.\n\nHenriques seems to claim to see a way how to circumvent the\nproblem discussed in that section, which consists of the\nfact that the 2-groups associated to the Lie 2-algebras\ncoming from the above L_2 algebra are not Lie.\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>On Tue, 1 Feb 2005, Thomas Larsson wrote:

> Urs Schreiber wrote:
>
> > Cook up any action principle which involves a non-associative operation at
> > one point or another. Quantize this action. The resulting quantum theory
> > will not involve non-associative quantum operators. Similarly, the
> > quantization of a nonlinear field theory will not involve non-linear quantum
> > operators.
>
> I'm not sure that I can cook up any action principle which involves a
> non-associative operation.

Let

(.,.)

R
(x,y)

be a non-associative product on the reals.
Let

x : R

be the trajectory of some non-relativistic particle, being
an element of some suitable function space F and consider
the action

S : F -> R
$LaTeX Code: x |-> \\int ( (dx(t)/dt)^2 - (x(t),x(t)) ) dt .$

It is as simple as that. An action is a map from the space of
configurations to the real numbers. The nature of the theory
obtained by taking the path integral using that action does
not depend on the choice of algorith by which you may want
to compute the number associated to a given configuration,
i.e. the algorithm by which you want to compute the value
of S on a given configuration x.

We had a similar discussion some time ago on spr. There you
were concerned about noncommutative field theories. In that case,
too, the noncommutativity of any product that appears in
the formula for computing the action does not affect the
properties of the product of the operators on the Hilbert
space of the quantum theory obtained from that action.

> > One example are certain flavors of string field theory, which do involve
> > non-associative operations in their action.
>
> Hm, hm. That ain't impressing me much,

It is an example that you were asking for.

> > Then, it is remarkable that Henriques' construction does not really involve
> > any non-associativity after all. The bracket
> >
> > [(X,a), (Y,b)]

> >
> > satisfies the Jacobi identity on the nose.
> >
> > The Jacobiator in this example degenerates to an automorphism 2-morphism on
> > the on 1-morphism [[(X,a),(Y,b)],(Z,c)].
>
> The Jacobi identity holds of course for the bilinear bracket, but how
> can then [(X,a),(Y,b),(Z,c)] != ?

Because the trinary bracket is not equal to the Jacobi operation on the
binary bracket in a 2-term $LaTeX Code: L_{oo}$ algebra. Instead, there is also a
unary operation d(.) and one of the relations to be satisfied by a
2-term $LaTeX Code: L_{oo}$ algebra (or

algebra') is

[x,[y,z]] + [[x,z],y] - [[x,y],z] .

where x stands for (x,0) etc.

Another relation is

[x,y,d(h)]

LaTeX Code: = [x,[y,d(h)]] + [[x,d(h)],y] - [[x,y],d(h)] .

where h stands for (0,h).

Comparing these two relations with the brackets that we are given
one concludes that d must act trivially by sending everything to 0,
because the right hand side of both these relations vanishes in
the present case. This is why I wrote in a previous email that

> where, BTW, I assume that Henriques means to set d(a) =

Indeed, Henriques didn't bother spelling this out because he is
considering a special example of the algebras discussed
in example 55 of math.

.

You can find the general definition of an

algebra on
p. 28 of that paper.

This has an intriguing relation to loop space calculus,
as explained in

Getzler & Jones, Ill. J. Math. 34,2 (1990), p. 256

and indeed in Hofman's paper, the central argument of which
I review in http://www-stud.uni-essen.de/~sb0264/2conn2.pdf .

The difference between the above

algebra and an ordinary
Lie algebra is rather small. Still, 'exponentiating' it
gives something interesting, as discussed in section 8.5 of
math.

.

Henriques seems to claim to see a way how to circumvent the
problem discussed in that section, which consists of the
fact that the 2-groups associated to the Lie 2-algebras
coming from the above

algebra are not Lie.