3.2.1 Induced Poisson structurei
In the sequel, we will use shuffle Hopf algebra language again. In order to do that, we have to catch up in notation and translate the concept of invariant charges, mostly following the accounts given in [Mei09] and [BM11] throughout this section again.
We have two multiplications on the shuffle algebra – the shuffle product (definition 1.1.2:2), which is commutative and associative, and the Pohlmeyer-Rehren Lie bracket (def-inition 1.4.3). We can now extend the Lie bracket derivatively to shuffle products of Euler elements, i.e.
e(x),e(y) := [e(x),e(y)], (3.7)
e(x)#e(y),e(z) := e(x)#{e(y),e(z)},+{e(x),e(z)}#e(y) (3.8) forx,y,z∈X∗. 5
5Cf. [Mei09, p. 50f]. There, it is not explicitly pointed out that this is well-defined because there are no multiplicative relations by theorem 1.2.2:4 – otherwise, it would have to be proved that rewriting the left hand side using such a relation could not lead to ambiguities.
Remark 3.2.1. Relations (1.33) and (1.34) imply that
Sh(X) = spanK(e(x1)#. . .#e(xk)|k∈N0,xi∈X∗∀i∈ {1, . . . ,k}) (3.9) as sets, and they can be used to calculate the Poisson brackets of two words. To calculate {x, y}for wordsx,yof lengthsnandmrespectively (cf. [Mei09, p. 51]),
1. rewritexandyin terms of sums of shuffle products of Euler elements using equation (1.33), and use linearity of the Poisson bracket, yielding
{x, y} =
2. use Leibniz’s rule to move all shuffle products out of the brackets (since # is commuta-tive, we can collect all factors on one side),
. . .= 3. evaluate the remaining brackets of the form{e(v),e(w)}, leaving us with a linear
combi-nation of shuffles of Euler elements
4. re-write all the Euler elements in terms of words using equation (1.34), so we obtain a linear combination of shuffles of words
5. and finally evaluate the shuffles, which leaves us with the desired linear combination of words.
Theorem 3.2.2. 1. (Sh(X),#, {·, ·})is a Poisson algebra, henceforth to be designatedi. 2. iis graded by thecombined degreedefined as
deg e(x1,1. . .x1,n1)#. . .#e(xk,1. . .xk,nk) Proof. 1. (Sh(X), #, {·, ·}) inherits the axioms for the Poisson bracket from the Lie algebra
g; compatibility with the shuffle product is by construction (cf. [Mei09][p. 50]).
2. can easily be proved by counting letters (this is [Mei09][Satz 4.8]).
3.2.2 Poisson algebra of invariant chargesh
ThePoisson algebra of invariant chargescan be constructed in the shuffle Hopf algebra language in an analogous way to equation (3.3).
Theorem 3.2.3(Poisson algebra of invariant charges). Let Z : Sh(X) → Sh(X) be the cyclic symmetrization map, i.e. for all words x1. . .xn∈X∗
Z(x1. . .xn) :=
n
X
i=1
xi. . .xnx1. . .xi−1 =Zx
1,...xn)inPohlmeyer’snotation
. (3.13) Then
h := imZ ⊂ i(= Sh(X)) (3.14)
is a Poisson subalgebra ofi(that inherits the gradation fromi) and is called thePoisson algebra of invariant charges.
Proof. This is [Mei09, Satz 4.9], based on [PR86, Proposition 12].
We have thus compiled a small dictionary of terms in Pohlmeyer’s original language and their analogues in the shuffle Hopf algebra language.
original term symbol shuffle Hopf language term symbol
tensor Rx
1,...,xn word x1. . .xn∈Xn
... with cyclically minimal indices Lyndon word
truncated tensor Rt
x1,...,xn Euler element e(x1. . .xn)
invariant charge Zx
1,...xn invariant charge Z(x1. . .xn) tensorial multiplication ·(or omitted) shuffle product # (or omitted) modified Poisson bracket [·, ·] Pohlmeyer-Rehren Lie bracket [·, ·]
Table 3.1: Corresponding terms in the languages used in Pohlmeyer’soriginal formulation and the reformulation in terms of the shuffle Hopf algebra due to Bahnsand Meinecke. Remark 3.2.4. For the same reason (lemma 2.2.2) thatgis ag0-module, the Poisson algebra of invariant chargeshis ah0-module, consideringh0as aLiesubalgebra6. It was shown that h0 so(d−1,C). Therefore, many of the features familiar fromgsuch as the decomposition into irreducible modules (multiplets) also occur in h. Note however, that for a given d, h0 so(d−1,C)so(d,C)g0.
3.2.3 The rest frame
A consequence [Poh99, p. 3] of definig equation (3.1) is that the Euler idempotents of single-letter words are the components of the totald-dimensional momentumP of the string. In fact, we can write
e(x) = x = Z
u(σ1, τ)dσ1 = Z
px(σ1, τ)dσ1 = Px ∀x∈X. (3.15)
6Note thath0is not aPoissonsubalgebra ofhsince the shuffle product is of degree+1 (cf. equation (3.12)).
We can now distinguish two cases depending on the Lorentz squareP2 :=νxyPxPyof the string. IfP2 =0, we are in themasslesscase. Ifm2 :=P2 > 0, we are in themassivecase and callmthe string’s invariant mass. Because the theory is much more developed for massive strings, we will only deal with this case.
Since the Poincaré group acts onh, leaving the combined degreel invariant, we can use a Lorentz transformation to the string’s relativistic rest frame, and in this situation,e(0)=m is the string’s invariant mass whilee(x)=0 for allx∈X\ {0}. A useful consequence of this is that in expressions of invariant charges as Euler-Lyndon words, many terms disappear, for instance
Z(e(2)(012)) = 2me(1,2).
We designate the Poisson algebra of invariants for the string’s rest frame byhm. Having this simplification at our disposal, we can now turn our attention to some statements about the structure ofhm.
3.2.4 Generation ofhm as a Poisson algebra Theorem 3.2.5(standard invariants). The elements ofhm
Z(e(2)(0ab)) ∈ h0m, 1
KZ(e(2)(0a0K−1b))) ∈ hKm−1, (K−1)!Z(e(K)(0K−1ax1. . .xK−1b)) ∈ hKm−1
for K>2, where a,b∈X\ {0}and xi ∈X for all i as well as xi ,0for at least one i,freelygenerate hmas an algebra (using shuffle multiplication). These generators are called thestandard invariants.
Proof. See [PR86, Proposition 17]
Unlike for the shuffle multiplication, there are relations between (multiple) Poisson brack-ets of the standard invariants. A large subalgebra ofhm, denoted byU, is the subalgebra generated – as a Poisson algebra – by the elements ofh0mandh1m. But not all elements ofhm are contained inU.
Definition 3.2.6 (exceptional element). An element of hlm with l ≥ 2 that is linearly inde-pendent from all (multiple) Poisson brackets of standard invariants of degree < lis called exceptional element.
The subspace spanned by the exceptional elements is infinite-dimensional:
Proposition 3.2.7. The invariants
Llg:= X
µ,ν
gµνZ(e(2)0µ0lν) (3.16) with l=2n+1, n∈N+are exceptional elements.
Proof. Pohlmeyerand Rehren’sproof (see [PR86, p. 622]) works by considering the leading
the right hand factor of which is an element of the Lie subalgebra
ker∂R0 ∩ker∂L0 =span(x∈X∗withx=1∅orx1=xn=0)
ofg, and showing that this term cannot be produced as a linear combination of leading terms
of other invariants.
All known exceptional elements could be modified by adding Poisson brackets of stan-dard invariants such that the resultingmodified exceptional elementscommute with each other.
This has lead to the following conjecture (cf. [Poh99, p. 8ff]):
Conjecture 3.2.8.
hm =anU, (3.17)
whereais an abelian Lie algebra spanned by modified exceptional elements.
Since all known exceptional elements as well as the generators ofU have homogeneity degree 2, an additional conjecture was reached.
Conjecture 3.2.9(Quadratic generation hypothesis). As Poisson algebras,hmas well ashare generated by their elements of homogeneity degree 2 (also called “quadratic elements”).
The quadratic generation hypothesis is not only interesting in its own right; it will be of great importance to the Rehren-Meinecke approach to string quantization that will be discussed in the next section.
Remark 3.2.10. Ford=3, all relations between elements ofanUup to degreel=5 have been computed [Hap93] by K. M. Happleusing computer algebra, applying Hall bases similar to the ones given in chapter 4.
More on Happle’swork and its relationship to the algorithms developed in chapter 4 can be found in remark 4.3.14.
Remark 3.2.11. The exceptional elementsLlggiven in equation (3.16) are not the only excep-tional elements. Pohlmeyer also pointed out in his last publication [Poh06, p.3] (without a proof) that an exceptional element with leading term