Hall sets

By construction of the free algebra,M(X) is a (K-linear) basis ofA(X). SinceL(X)=A(X)/ILie, M(X) spansL(X) as well, but due to the relations introduced by the idealI_Lie, the setM(X) is not linearly independent inL(X).

Fortunately, the concept ofHall setsprovides us with a family of subsets ofπ(M(X)) each of which is linearly independent but still spansL(X).

Definition 4.2.6(Hall set, Hall order). LetXbe a set and let<be a total order onM(X). We define the setH⊂M(X) by

1. X⊂H,

2. ifh₁,h₂∈M(X) then (h₁,h₂)∈Hiffh₁,h₂∈Hand (a) h₁<h₂ ,

(b) h₂∈Xorh₂ =(a,b) witha≤h₁.

If the order<satisfies (h₁,h₂)>h₂for allh₁,h₂∈Hsuch that (h₁,h₂)∈H, the order<is called aHall orderandH(which is uniquely determined by<) is called theHall setcorresponding to<.³

In chapter 1 we had introduced the concept of the word algebra, which is just another term for the freeassociative(but non-commutative) algebraKhXiover an alphabetX, spanned by the set of wordsX^∗. It is not very surprising that there is a canonical connection⁴to the concept of the freenon-associativealgebraA(X) over the same alphabet spanned by the set of monomialsM(X).

3Note that the order>is inverted compared to the notation in [dG00], but consistent with the notation in [Ser65, p. 22]. This is done to maintain notational consistency with the degree of the Pohlmeyer-Rehren Lie algebra as well as the more intuitive convention of (h1,h2)>h2instead of (h1,h2)<h2as a condition for a Hall order. Several conventions for Hall sets that switchh₁andh₂oraandbexist, and due to the anticommutativity of the Lie bracket, they are all equivalent in the sense that theorem 4.2.12 holds.

4Equivalently to the following definition (and similar to the definition of the free Lie algebra 4.2.3), the free associative algebraKhXican be defined as the quotient algebraA(X)/IwhereIis the ideal generated by elements of the form (a,(b,c))−((a,b),c) fora,b,c∈M(X), and the extension ofφas the canonical projection.

Definition 4.2.7(foliage, Hall word). Consider the mapϕ:M(X)→X^∗⊂KhXicalledfoliage defined by

ϕ(x) :=

(x ifx∈X

ϕ(x⁰)_ ϕ(x”) ifx=(x⁰,x”)∈M(X)\X . (4.2)

Simply put, the foliage removes the non-associative order in which monomials are composed, turning them into (associative) words.

IfHis a Hall set relative to<H andh∈H, thenϕ(h) is called aHall word(relative to<H).

The foliageϕcan be linearly extended to a mapA(X)→KhXi.

We continue our general discussion of Hall sets with three basic examples of Hall orders.

Example 4.2.8(some Hall orders). Letm,n∈M(X).

1. The foliageϕcan be used to extend any order defined onX^∗toM(X). Some of them are Hall orders. It is straightforward to show (cf. [dG00, lemma 7.8.4]) that thelexicographic order, defined by

m<Lex n :⇔ϕ(m)<Lexϕ(n) (4.3) is a Hall order.

2. Analogously thegraded lexicographic orderis extended toM(X):

m<DegLexn :⇔ϕ(m)<DegLex ϕ(n). (4.4)

3. <DegLex is a special case of the following fact that is easy to demonstrate as well: Any order>onM(X) that satisfies

deg_Hm>deg_Hn⇒m>n (4.5) for allm,ninM(X) is a Hall order.

It is a useful fact that the set of Lyndon words, introduced in section 1.3, can be considered a particular Hall set.

Theorem 4.2.9 (Lyndon words as a Hall basis). The Hall words relative to <Lex are exactly the Lyndon words. ϕ|_H is injective.

Proof. See [dG00, Chapter 7]. DeGraafdefinesϕ(H) as the set of Lyndon words and proves lemma 7.8.5, which states that they can be characterized by the property used to define them in definition 1.3.2. He also gives an algorithm to compute the corresponding Hall element to

a Hall word.

Since ϕ|_H is a bijection between elements of the Hall set and Hall words, we will use the terms synonymously, as the literature also frequently does. Its inverse ϕ|⁻¹

H is called bracketing.

Remark 4.2.10. The fact that ϕ|_H is bijective gives rise to an interesting question for the Pohlmeyer-Rehren Lie algebra: If the Euler-Lyndon words are Hall words, can we find generatorsx₁, . . . ,x_d ∈ g such that the Lyndon word (y₁. . .y_n) in any Euler-Lyndon word e(y₁. . .y_n) corresponds to a way to generate that Euler-Lyndon word in the following way:

e(y₁. . .y_n) = zϕ|⁻¹

H (x_y1. . .x_yn) withz∈C? (4.6)

The right hand side of this equation is a multiple Lie bracket of the elementsx_y1. . .x_ynwhere the order is controlled by the property that it is a Hall word.

We begin exploring this question ford=3. We also postulate thatx_iare compatible with the magnetic quantum numbers in the following way: x₊ ∈ g¹, x₀ ∈ g⁰, x− ∈ g⁻¹. If this is the case, the magnetic quantum numbers on both sides of equation (4.6) automatically match due to additivity of the magnetic quantum numbers (lemma 1.6.4). Otherwise, for the same reason, the multiple Lie brackets on the right hand side have summands with magnetic quantum numbers different from the left hand side that all have to vanish foreveryHall word on the left hand side.

Note that forn=2, satisfying equation (4.6) is possible with x₀ = e(−+), x± = e(0±)

(a calculation that also shows that ford=3,g₀is isomorphic tosl₂by giving us the structure constants).

However, forn=1; thexiare fixed by equation (4.6) to be (up to complex constants) x₀ = e(0), x± = e(±),

but these elements do not satisfy equation (4.6) for anyn>1 because they are in the center ofg so the right hand side would be zero. So, the initial question can be answered in the negative.

But what if we exclude the central subalgebrag⁻¹, in other words also presupposen≥2?

In general, we can not satisfy equation (4.6) for allnwith homogeneousx_i (with respect to the usual gradation deg ong, cf. thm. 1.4.2:4) because then

degx₀+2 degx₊ =dege(0++) = 1 =dege(00+) = 2 degx₀+degx₊

which is impossible to satisfy withx_i ∈ N0. But as the following proposition shows, even inhomogeneousx_icannot satisfy equation (4.6) for alln≥2.

Proposition 4.2.11. There are no x₀∈g⁰, x±∈g^±1such that equation(4.6)holds for all n≥2.

Proof. We writex₀,x₊in terms of the basis ofg₀andg₁lined out in subsection 2.3.3:

x₊ = a e + b¹v¹ + c²v¹ + u, (4.7) x₀ = i2h + j¹v⁰ + k²v⁰ + w (4.8) with coeffcientsa,b,c,i,j,k ∈ Cand remaining termsu,w ∈ g≥2. Postulate (4.6) implies the equations (the complex coefficientszcan be absorbed into thea,b,c,i,j,k)

[x₀,x₊] = e(0+), (4.9)

[x₊,[x₀,x₊]] = e(0++). (4.10)

Substituting the values forx₀ andx₊from equation (4.9) and comparing the coefficients of the Euler-Lyndon basis for degrees 0 and 1 (noting that all terms withuorvare of degree≥ 2), we obtain a system of equations which has the unique solution

a=i=1 andb=c= j=k=0. (4.11) Analogously, from equation (4.10), we can deduce

ak²=1, (4.12)

in contradiction to the solution 4.11 of equation (4.9).

Now that we have established Hall sets, it is time to introduce a very important application of Hall bases that is eminently useful in the theory of free Lie algebras and also is the foundation of the entire chapter.

Theorem 4.2.12. Let H be a Hall set in M(X). Then H is a (K-linear) basis of the free Lie algebra L(X).

Proof. This is [dG00, Corollary 7.7.7].

In this situation, we refer toH as aHall basisof L(X). An intuitive way to think about the Hall set in light of this theorem is that condition 2a (for the Hall set, definition 4.2.6) eliminates elements ofM(X) that are linerarly dependent because of the alternativity of the Lie bracket and condition 2b eliminates elements ofM(X) that are linearly dependent because of its Jacobi relation.

Before an algorithm providing a way to list the elements of a Hall set for practical computations is given, we apply the previous two theorems to prove a classical result from the theory of free Lie algebras originally proved by Wittin 1937 [Wit37][p. 152f]. It allows us to calculate the dimension of the strata (with respect to the monomial degree) of the free Lie algebra in terms of theorem 1.3.6.

Corollary 4.2.13(Witt’s formula). Let X be a finite set. Then dimL(X)_l = NumLyndon(l,#X) = 1

l X

q|l

µ(q) (#X)^l/q. (4.13)

Proof. The dimension ofL(X)_lis independent of the choice of basis. The Hall set relative to<lex

is a basis by theorem 4.2.12, and by theorem 4.2.9, the Hall words of degreelcan be identified with the Lyndon words of lengthl, the number of which is given by NumLyndon(l,#X).