The Lie Algebra of the Pure Braid Group

Miguel A. Xicoténcatl

Abstract

We study the Lie algebra E_{∗}^{0}(P_{k}) associated to the descending central series of the pure braid group P_{k}, and prove that is a “twisted” extension of free Lie algebras. The “twisting” is given by the infinitesimal braid relations.

1991 Mathematics Subject Classification: 17B01, 20F14, 20F36, 20F40. Keywords and phrases: Free Lie algebra, pure braid group, descending central series, configuration spaces.

1 Introduction

In this article we describe the Lie algebra E_{∗}^{0}(P_{k}) associated to the descending central series of the pure braid group P_{k}. The additive structure of E_{∗}^{0}(P_{k}) was computed in [F-R] by M. Falk and R. Randell. The Lie algebra structure was first found by T. Kohno [Ko85] using rational homotopy and mixed Hodge theory, and apart from the grading, it is now know to be ismorphic to the Lie algebra of primitives of H_{∗}(ΩF (ℝ^{2n}, k) for n ≥ 2. In this paper, the structure of the Lie algebra is worked out directly using a presentation of the braid group.

The integral homology of ΩF (ℝ^{2n}, k) has been computed by E. Fadell and S. Husseini [F-H] and subsequently by F. Cohen and S. Gitler [C-G], who showed that the Lie algebra of primitives is given by a non-trivial extension of free Lie algebras. Namely, there is an isomorphism from the module of primitives P H_{∗}(ΩF (ℝ^{2n}, q)) ≅ L_{2} ⊕ … ⊕ L_{q}, where L_{i} = L[A_{i,1}, …, A_{i,i−1}] is a free Lie algebra on i − 1 generators of degree 2n − 2, and the relations among the A_{i,j}’s are of the form:

\begin{aligned}[A_{i,j}, A_{k,i}] & = & [A_{k,i}, A_{k,j}] \\ [A_{i,j}, A_{k,j}] & = & [A_{k,j}, A_{k,i}] \end{aligned}  for 1 ≤ j < i < k ≤ q

and [A_{i,j}, A_{k,l}] = 0 for distinct i, j, k, l. These are known as the “infinitesimal braid relations” or “Yang-Baxter relations” and they have appeared independently in several contexts in algebra and topology (See for example [Ko87]).

A satisfactory explanation for the connection between both Lie Algebras is still missing, although this appears to be part of a more general phenomenon. Namely, the work D. Cohen [DC] and of the author [Xi97] proves the existence of analogous isomorphisms for certain choices of “orbit configuration spaces” of points in ℂ^{m} − {0}, acted by ℤ/n. Similar results were obtained by F. Cohen and the author [C-X] for the case of the orbit configuration space of points in ℂ^{m} under the action of a lattice.

The author wishes to thank the referees for some useful remarks and suggestions. In particular, for pointing out Kohno’s orginal proof and the need of an argument to show that the infinistesimal braid relations are enough to describe the Lie algebra extension.

2 The Lie algebra E_{∗}^{0}(G)

Recall that given a group G, its descending central series G = Γ_{1} ≥ Γ_{2} ≥ … is defined as: Γ_{1} = G, Γ_{n} = [Γ_{n−1}, G]. Notice that Γ_{n+1} is normal in Γ_{n} and so we can form the associated graded group of this filtration {E_{n}^{0}}_{n≥1} by setting E_{n}^{0}(G) = Γ_{n}/Γ_{n+1}, which is an abelian group for all n. Put E_{∗}^{0}(G) = ⨁_{n} E_{n}^{0}(G). Now for every p and q the commutators [a, b] = aba^{−1}b^{−1} in G induce a bilinear map

E_{p}^{0} × E_{q}^{0} ⟶ E_{p+q}^{0}

which makes E_{∗}^{0}(G) into a Lie algebra. Moreover, E_{∗}^{0} is a functor from the category of groups to the category of Lie algebras. Two elementary properties of E_{∗}^{0} are:

1. E_{1}^{0}(G) = H_{1}(G; ℤ) (with ℤ-trivial coefficients).

2. If F [S] is the free group generated by the set S, then E_{∗}^{0}(F [S]) is isomorphic to L[S], the free Lie algebra generated by S (see [Serre]).

In [F-R] Falk and Randell have proven Lemma 2.1 Let 1 → A →^{i} B →^{j} C → 1 be a split extension of groups such that C acts trivially on H_{1}(A). Then, the induced sequence

0 → E_{∗}^{0}(A) → E_{∗}^{0}(B) → E_{∗}^{0}(C) → 0

is split exact (as graded abelian groups, but not necessarily as Lie algebras).

Sketch of the proof. Let σ : C → B be a cross-section. There is a well defined map τ : B → A given by τ(b) = i^{−1}[σj(b^{−1}) · b] which need not be a homomorphism. Now, the triviality of the action is equivalent to [A, C] ⊂ [A, A] = Γ_{2}A, where we identify A and C with their images under i and σ. One can prove inductively that [Γ_{n}A, Γ_{k}C] ⊂ Γ_{n+k}A and use this to show that τ(Γ_{n}B) ⊂ Γ_{n}A. Now one can show that the induced sequence

1 → Γ_{n}A → Γ_{n}B → Γ_{n}C → 1

is (split) exact, since b ∈ Γ_{n}B ∩ ker j implies b = i(τ(b)). This is, τ is a right inverse of the restriction Γ_{n}A →^{i} Γ_{n}B for every n. The theorem follows easily from this last statement. □

Thus, there is an isomorphism E_{∗}^{0}(B) ≅ E_{∗}^{0}(A) ⊕ E_{∗}^{0}(C) of abelian groups. In general, this is not a trivial extension of Lie algebras but under the same hypothesis we can prove that if a ∈ E_{∗}^{0}(A) and c ∈ E_{∗}^{0}(C) then [a, c] ∈ E_{∗}^{0}(A).

Remark. Notice that E_{∗}^{0}(A) and E_{∗}^{0}(C) inject into E_{∗}^{0}(B) as Lie algebras. This follows at once from the functorial properties of E_{∗}^{0}, since the induced morphisms

i_{∗} : E_{∗}^{0}(A) ⟶ E_{∗}^{0}(B)
σ_{∗} : E_{∗}^{0}(C) ⟶ E_{∗}^{0}(B)

are monomorphisms of Lie algebras.

3 The pure braid group

Given a manifold M, let M^{k} be the k-fold cartesian product of with itself M^{k} = {(x_{1}, …, x_{k}) | x_{i} ∈ M}.

Definition 3.1 The configuration space F (M, k) of k-ordered tuples of points in M is given by:

F (M, k) = {(x_{1}, …, x_{k}) ∈ M^{k} | x_{i} ≠ x_{j} if i ≠ j}.

Recall that the configuration space F (ℝ^{2}, k) is an Eilenberg-Mac Lane space of type K(π, 1), and its fundamental group can be identified with the pure braid group on k strands, P_{k}. From [F-N] and [CLM] we know that there is a fibration:

(ℝ^{2} − Q_{k−1}) ⟶ F (ℝ^{2}, k) ⟶ F (ℝ^{2}, k − 1)

which has a cross-section and trivial local coefficients in homology. Thus by looking at the homotopy exact sequence, we obtain the split extension

1 ⟶ F_{k−1} ⟶ P_{k} ⟶ P_{k−1} ⟶ 1

which satisfies the hypothesis of the lemma above. Here F_{k−1} is a free group on (k−1) generators, usually denoted A_{k,1}, A_{k,2}, …, A_{k,k−1}. Applying inductively Lemma 2.1 we get

E_{∗}^{0}(P_{k}) ≅ E_{∗}^{0}(F_{1}) ⊕ E_{∗}^{0}(F_{2}) ⊕ … ⊕ E_{∗}^{0}(F_{k−1})
≅ L_{2} ⊕ L_{3} ⊕ … ⊕ L_{k}

with L_{i} = L[A_{i,1}, …, A_{i,i−1}], a free Lie algebra on (i − 1) generators. Also, notice that each L_{i} injects into E_{∗}^{0} as a Lie algebra.

Thus, to work out the Lie algebra extension it suffices to give the products:

[ , ] : L_{j} ⊕ L_{i} ⟶ L_{i} for i > j

Let B_{i} and B_{j} be bases for L_{i} and L_{j}, respectively, considered as free abelian groups (take for instance a Hall basis for each free Lie algebra [Serre]). Since the operation is bilinear, we can restrict ourselves to calculate the products [x, y] for x ∈ B_{j} and y ∈ B_{i}. Furthermore, the Jacobi identity reduces the calculation to the knowledge of the Lie brackets among the generators of L_{j} and L_{i}. Finally, we can use a presentation of P_{k} and obtain the infinitesimal braid relations among the A_{i,j}’s. Then we have

Theorem 3.2 The Lie algebra E_{∗}^{0}(P_{k}) associated to the D.C.S. of the pure braid group P_{k} is given as a module by L_{2} ⊕ L_{3} ⊕ … ⊕ L_{k} where L_{i} = L[A_{i,1}, …, A_{i,i−1}] is the free Lie algebra generated by A_{i,1}, …, A_{i,i−1} . Moreover, the inclusion L_{i} ↪ E_{∗}^{0}(P_{k}) is a homomorphism of Lie algebras and the Lie brackets among the generators are given by:

(a) If {i, j} ∩ {r, s} = ø then [A_{i,j}, A_{r,s}] = 0. Otherwise

(b) For j < i < r, we have: [A_{i,j}, A_{r,i}] = [A_{r,i}, A_{r,j}] and [A_{i,j}, A_{r,j}] = [A_{r,j}, A_{r,i}].

4 Lie relations on E_{∗}^{0}(P_{ℓ})

Since P_{ℓ} is isomorphic to π_{1}(F (ℝ^{2}, ℓ)) then H_{1}(F (ℝ^{2}, ℓ)) = P_{ℓ}/[P_{ℓ}, P_{ℓ}] = E_{1}^{0}(P_{ℓ}). Remember that F (ℝ^{2}, ℓ) is acted by the symmetric group Σ_{ℓ}. Therefore every element σ of Σ_{ℓ} is regarded as homeomorphism

σ : F (ℝ^{2}, ℓ) ⟶ F (ℝ^{2}, ℓ)

and particular, if 1 ≤ j < i < k ≤ ℓ, we can choose σ to be the permutation which is the product of the transpositions (1, j), (2, i) and (3, k). For such σ, the induced map in homology interchanges

A_{i,j} ⟼ A_{2,1}
A_{k,j} ⟼ A_{3,1}
A_{k,i} ⟼ A_{3,2}

Now, recall from [F-N] that the projection onto the first 3 coordinates, F (ℝ^{2}, ℓ) → F (ℝ^{2}, 3), is a fibration that maps A_{i,j} ↦ A_{i,j} for 1 ≤ j < i ≤ 3, and that it has a cross-section.

Thus, to compute the Lie brackets: [A_{i,j}, A_{k,i}] and [A_{i,j}, A_{k,j}] for 1 ≤ j < i < k ≤ ℓ, it enough to to prove the following relations

[A_{2,1}, A_{3,2}] = [A_{3,2}, A_{3,1}]  (1)
[A_{2,1}, A_{3,1}] = [A_{3,1}, A_{3,2}]  (2)

in Γ_{2}/Γ_{3} and the rest follow by naturality.

Remember that P_{ℓ} is a normal subgroup of the braid group B_{ℓ} and that the last can be presented as the quotient of the free group generated by σ_{1}, σ_{2}, …, σ_{ℓ−1} subject to the relations

σ_{i}σ_{j} = σ_{j}σ_{i} if |i − j| ≥ 2
σ_{i}σ_{i+1}σ_{i} = σ_{i+1}σ_{i}σ_{i+1} for i = 1, …, ℓ − 2

It is known (see for example [Birman] pg. 20) that we can take the elements

A_{i,j} = σ_{i−1}σ_{i−2} … σ_{j+1}σ_{j}^{2}σ_{j+1}^{−1} … σ_{i−2}^{−1}σ_{i−1}^{−1}

to be the generators of P_{ℓ} for 1 ≤ j < i ≤ k. So when ℓ = 3, a presentation for B_{3} is

B_{3} = ⟨ a, b | aba = bab ⟩

where a = σ_{1} and b = σ_{2}. We also have

A_{2,1} = a^{2}
A_{3,1} = ba^{2}b^{−1}
A_{3,2} = b^{2}

First, notice that

A_{2,1}A_{3,1}A_{2,1}^{−1} = a^{2}ba^{2}b^{−1}a^{−2}

A_{3,2}^{−1}A_{3,1}A_{3,2} = b^{−2}(ba^{2}b^{−1})b^{2}
= b^{−1}a^{2}b

In the other hand we have that

b = bab^{−1}(bab)b^{−1}a^{−2}

= bab^{−1}(aba)b^{−1}a^{−2}

= bab^{−2}(bab)ab^{−1}a^{−2}

= bab^{−2}(aba)ab^{−1}a^{−2}

= bab^{−2}aba^{2}b^{−1}a^{−2}

Then we have

b^{−1}a^{2}b = b^{−1}a^{2}(bab^{−2}aba^{2}b^{−1}a^{−2})

= b^{−1}a(aba)b^{−2}(aba^{2}b^{−1}a^{−2})

= b^{−1}a(bab)b^{−2}(aba^{2}b^{−1}a^{−2})

= b^{−1}(aba)b^{−1}(aba^{2}b^{−1}a^{−2})

= b^{−1}(bab)b^{−1}(aba^{2}b^{−1}a^{−2})

= a^{2}ba^{2}b^{−1}a^{−2} From the last equation we observe that

A_{2,1}A_{3,1}A_{2,1}^{−1} = A_{3,2}^{−1}A_{3,1}A_{3,2}

and thus

[A_{2,1}, A_{3,1}] = [A_{3,2}^{−1}, A_{3,1}]

In the associated graded E_{∗}(P_{3}) this means

[A_{3,2}^{−1}, A_{3,1}] = [A_{3,1}, A_{3,2}^{−1}]^{−1}

= −[A_{3,1}, −A_{3,2}] = [A_{3,1}, A_{3,2}]

and the relation (2) above follows. For (1) start by noticing that

[A_{3,2}, A_{2,1}] = b^{2}a^{2}b^{−2}a^{−2}
[A_{3,1}^{−1}, A_{3,2}^{−1}] = (ba^{−2}b^{−1})b^{−2}(ba^{2}b^{−1})b^{2}
= ba^{−2}b^{−2}a^{2}b

Also, notice that

a = b^{−1}(bab)b^{−1}
= b^{−1}(aba)b^{−1}
= b^{−1}a(bab)b^{−2}
= b^{−1}a(aba)b^{−2}
= b^{−1}a^{2}bab^{−2}

and therefore:

a^{2} = a(b^{−1}a^{2}bab^{−2})  or
1 = ab^{−1}a^{2}bab^{−2}a^{−2}

and then we have :

a^{−2}b^{−2}a^{2}b = a^{−2}b^{−2}a^{2}b(ab^{−1}a^{2}bab^{−2}a^{−2})

= a^{−2}b^{−2}a(aba)b^{−1}a^{2}bab^{−2}a^{−2}

= a^{−2}b^{−2}a(bab)b^{−1}a^{2}bab^{−2}a^{−2}

= a^{−2}b^{−2}(aba)a^{2}bab^{−2}a^{−2}

= a^{−2}b^{−2}(bab)a^{2}bab^{−2}a^{−2}

= a^{−2}b^{−1}(aba)abab^{−2}a^{−2}

= a^{−2}b^{−1}(bab)abab^{−2}a^{−2}

= a^{−2}(aba)bab^{−2}a^{−2}

= a^{−1}(bab)ab^{−2}a^{−2}

= a^{−1}(aba)ab^{−2}a^{−2}

= ba^{2}b^{−2}a^{−2}

Multiplying on the left by b we get:

b(a^{−2}b^{−2}a^{2}b) = b^{2}a^{2}b^{−2}a^{−2}.

So

[A_{3,2}, A_{2,1}] = [A_{3,1}^{−1}, A_{3,2}^{−1}]
= [A_{3,1}, A_{3,2}]

which gives relation (1). □

References

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[DC] D.C. Cohen, Monodromy of fibre-type arrangements and orbit configuration spaces, to appear in Forum Mathematicum.

[C-G] F.R. Cohen, S. Gitler, On loop spaces of configuration spaces, submitted to the Memoirs of the AMS.

[CLM] F.R. Cohen, T.J. Lada and J.P. May, The homology of iterated loop spaces, Lecture Notes in Math., vol. 533, Springer-Verlag (1976).

[C-X] F.R. Cohen and M.A. Xicoténcatl, On orbit configuration spaces associated to the Gaussian integers: Homotopy and homology groups, submitted to Topology and its Applications.

[F-H] E. Fadell and S. Husseini, The space of loops on configuration spaces and the Majer-Terracini index,Topol. Methods Nonlinear Anal. 11 (1998), 249-271.

[F-N] E. Fadell and L. Neuwirth, Configuration spaces, Math. Scand., 10 (1962), 111-118.

[F-R] M. Falk and R. Randell, The lower central series of a fiber-type arrangement, Inventiones Math. 82 (1985), 77-88.

[Ko85] T. Kohno, Série de Poincaré-Kozul associée aux groupes de tresses pures, Inventiones Math. 82 (1985), 57-75.

[Ko87] T. Kohno, Linear representations of braid groups and classical Yang-Baxter equations, Contemporary Mathematics vol. 78 (1987), 339-363.

[MKS] W. Magnus, A. Karras and D. Solitar, Combinatorial group theory, Dover (1976).

[Serre] J.P. Serre, Lie algebras and Lie groups, W.A. Benjamin, Inc. (1965).

[Xi97] M.A. Xicoténcatl, Ph.D. Thesis, The University of Rochester (1997).

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