Before we begin constructing the Pohlmeyer-Rehren Lie algebra, let us recall some very basic definitions and facts for easy reference.
Definition 1.4.1. (Lie algebra, direct sum, simple, semisimple, Poisson algebra, derived series, solvable, lower central series, nilpotent, Poisson algebra, generation)
1. LetGbe vector space over a fieldK. AK-bilinear map [·, ·] : G×G→Gis called aLie bracketif
[x, x] =0 (alternativity) and (1.46) [x,[y,z]] + [y,[z, x]] + [z, [x, y]] =0 (Jacobi identity) (1.47) are satisfied for allx,y,z∈G. Then (G,[·,·]) (or more concisely onlyG, if the Lie bracket is unambiguous) is called a Lie algebra (over K). In particular, any Lie algebra is an algebra and all terms defined for algebras such as the ones from definition 1.1.6 apply.
Because of equation (1.46), [x,y] =−[y,x] for allx,y ∈ Gand abelian Lie algebras are those with thetrivial Lie bracket0.
2. Thedirect sum of Lie algebras (G,[·, ·]G) and (H,[·,·]H) is defined as the vector space G⊕Hwith the Lie bracket [·, ·] :G⊕H×G⊕H→G⊕Hdefined by
[(g,h),(g0,h0)] := ([g, g0]G,[h,h0]H) ∀g,g0 ∈G,h,h0 ∈H. (1.48) 3. A Lie algebraGthat has no ideals except 0 andGitself is calledsimple. A Lie algebra
that is a direct sum of simple Lie algebras is calledsemisimple.
4. Let (G,[·, ·]) be a Lie algebra. Iteratively define two series of ideals, thederived series G(0) := G, G(n+1) := [G(n),G(n)]∀n∈N0 (1.49) ofGand thelower central series
G0 := G, Gn+1 := [G,Gn]∀n∈N0 (1.50) ofG. IfG(n) =0 for somen∈ N0, thenGis calledsolvable. IfGn= 0 for somen∈ N0, thenGis callednilpotent. From the definition, one concludes that nilpotent Lie algebras are solvable.
5. Let (G,{·, ·}) be a Lie algebra and let (G,·) be an associative algebra. If
{x·y, z} = x· {y,z} + {x, z} ·yfor allx,y,z∈G, (1.51) then (G,{·, ·}, ·) is called aPoisson algebra and the Lie bracket {·, ·}is called a Poisson bracket.
6. LetGbe a (Lie) algebra/Poisson algebra/group/ideal with a subsetX⊂ G. ThenG is calledgenerated(as a (Lie) algebra/Poisson algebra/group/ideal) byXif no subset H$Gexists such thatX⊂HandHis a (Lie) algebra/Poisson algebra/group/ideal.
If a finite setXexists such thatXgeneratesG, thenGis calledfinitely generated.
ThePohlmeyer-RehrenLie algebra and its structure constitute a major focus of this thesis.
It can be constructed from the shuffle algebra Sh(X) by equipping it with some additional structure. We begin by defining two closely related3derivations of Sh(X) and then use them to define a Lie bracket on im(e).
3. In terms of Euler elements, this Lie bracket can be written as
e(x1. . .xn),e(y1. . .ym)
2. This is can be demonstrated using 3: Alternativity then is a consequence of proposition 1.2.5, and the Jacobi identity can be proved by tracking the contributions of the cyclic product in the Jacobi identity according to thega,boccurring, observing that they cancel for certain terms, and then using 1.2.5 to rewrite all other contributions into such terms (see [BM11, Prop. 3] for details).
3. We evaluate the convolution product (notice the index shift forj):
[e(x), e(y)] = X
4. This is a simple counting argument: The word on the right hand side of equation (1.54) has two fewer letters than the words on the words on the left hand side have combined.
Note that the Lie bracket can also be understood as a sum over all possibilities which letters ofxto move to the right and of yto move to the left using Pohlmeyerand Rehren’s proposition 7 (1.2.5), then deleting the moved letters with the derivations∂Ra and∂La, weighing the summand with the entry ofgcorresponding to the deleted letters and concatenating the results.
Definition 1.4.3(Pohlmeyer-Rehren Lie algebra). If gis proportional to the metric tensorη of the Minkowski metric, i.e.
g=α η=αDiag(−1,1, . . . ,| {z }1
d−1
) (1.58)
withα∈C\{0}(where Diag denotes a diagonal matrix, its arguments being the entries of the diagonal), we callg:= (im(e), [·, ·]) thePohlmeyer-Rehren Lie algebra.
For reasons that will be discussed later in remark 3.1.2 the casesd =3 andd =4 will be of particular interest. We will later see (in section 1.8) how a decomposition ofgcan be used to bring to bear some combinatorial arguments on the words involved.
Remark 1.4.4. Note that the lowest nonzero stratum ofgis notl=0 butl=−1. Sinceg−1is central by the defining equation of the Lie bracket (1.54), it is often not considered explicitly.
In this sense,gcan be thought of as aN0-graded Lie algebra with added central elements instead of aZ-graded Lie algebra in which all strata of degree less than−1 are zero.
Remark 1.4.5. Since the Pohlmeyer-Rehren Lie algebra is the primary focus of this thesis, its history warrants some attention. It was first described by Pohlmeyerand Rehren([PR86]).
There, the Lie bracket was called the “modified Poisson bracket”, and Lie algebra was not explicitly named, but its elements were called “truncated tensors”. The context in which these notions were developed was Pohlmeyer’s approach to the quantization of the Nambu -Gotostring. The truncated tensors were defined by way of path-ordered integrals of the so calledleftandright movers, which are tangent vectors on the string’s world surface. Many of the facts used in this thesis were proved in that context.
The considerably simpler definition used here is taken from Bahnsand Meinecke, who proved it to be equivalent to Pohlmeyer’s and Rehren’s definition ([BM11, ]). Since it was not the focus of their work, they simply call it the “auxiliary Lie algebra”, but since its structure is the main subject of this thesis and the term “auxiliary Lie algebra” will be used for an unrelated object in the context of Kac-Moody algebras later, it deserves its proper name, at least within this scope.
A more detailed account of the backdrop of the quantization of the Nambu-Goto string in whichgwas discovered and the subsequent constructions in which it is featured is given in chapter 3.
Remark 1.4.6. Using any other diagonal matrix with complex nonzero entries instead ofα η in definition 1.4.3 ofgleads to an isomorphic Lie algebra; in this case the basis vectors can be multiplied with appropriate complex numbers to obtain the same structure constants. In Pohlmeyer’soriginal work recounted in chapter 3, the Minkowski metric is inherited from the physical problem of string quantization where it is the metric ofd-dimensional spacetime (with one time dimension). We continue this special treatment of the time dimension by using the Minkowski metric in the definition ofgas well as some other special considerations of the letter 0 in our alphabet later.
In Pohlmeyer’soriginal workα=2 is used due to the way the Lie bracket is constructed there, but since multiplying the structure constants of a Lie algebra with a global nonzero factor is an isomorphism of Lie algebras, this is not relevant to the structure ofg.