Symmetry, Integrability and Geometry: Methods and Applications
Vol. 2 (2006), Paper 080, 3 pages
A Formula for the Logarithm of the KZ Associator?
Benjamin ENRIQUEZ
†
and Fabio GAVARINI
‡
†
IRMA (CNRS), rue René Descartes, F-67084 Strasbourg, France
E-mail: enriquez@math.u-strasbg.fr
‡
Universitá degli Studi di Roma “Tor Vergata”, Dipartimento di Matematica,
Via della Ricerca Scientif ica 1, I-00133 Rome, Italy
E-mail: gavarini@mat.uniroma2.it
Received October 03, 2006, in final form November 10, 2006; Published online November 13, 2006
Original article is available at http://www.emis.de/journals/SIGMA/2006/Paper080/
Abstract. We prove that the logarithm of a group-like element in a free algebra coincides
with its image by a certain linear map. We use this result and the formula of Le and
Murakami for the Knizhnik–Zamolodchikov (KZ) associator Φ to derive a formula for log(Φ)
in terms of MZV’s (multiple zeta values).
Key words: free Lie algebras; Campbell–Baker–Hausdorff series, Knizhnik–Zamolodchikov
associator
2000 Mathematics Subject Classification: 17B01; 81R50
To the memory of Vadim Kuznetsov.
1
Logarithms of group-like elements
Let Fn be the free associative algebra generated by free variables x1 , . . . , xn , let fn ⊂ Fn be the
free Lie algebra with the same generators, and let bfn , Fbn be their degree completions (where
x1 , . . . , xn have degree 1). A group-like element of Fbn is an element of the form X = 1+ (terms of
degree > 0), such that ∆(X) = X ⊗ X, where ∆ is the completion of the coproduct Fn → Fn⊗2 ,
for which x1 , . . . , xn are primitive. It is well-known that the exponential defines a bijection
exp: bfn → {group-like elements of Fbn } (also denoted x 7→ ex ). We denote by log the inverse
bijection.
We denote by CBHn (x1 , . . . , xn ) the multilinear part (in x1 , . . . , xn ) of log(ex1 · · · exn ). Define
a linear map
cbhn : Fn → fn
by cbhn (1) = 0 and cbhn (xi1 · · · xik ) := CBHk (xi1 , . . . , xik ). This map extends to a linear map
d n : Fbn → bfn .
cbh
d n (X).
Proposition 1. If X ∈ Fbn is group-like, then log(X) = cbh
Proof . It is known that Fn = U (fn ), so that the symmetrization is an isomorphism
sym: S(fn ) → Fn . Denote by pn : Fn → fn the composition of sym−1 with the projection
S(fn ) = ⊕k≥0 S k (fn ) onto S 1 (fn ) = fn . We first prove:
Lemma 1. pn = cbhn .
?
This paper is a contribution to the Vadim Kuznetsov Memorial Issue “Integrable Systems and Related Topics”.
The full collection is available at http://www.emis.de/journals/SIGMA/kuznetsov.html
2
B. Enriquez and F. Gavarini
Proof . If g is a Lie algebra, let pg : U (g) → g be the composition of the inverse of the symmetrization S(g) → U (g) with the projection onto the first component of S(g). If φ: g → h is
a Lie algebra morphism, then we have a commutative diagram
pg
U (g) →
g
U (φ)↓
↓φ
ph
U (h) → h
It follows from Lemma 3.10 of [4] that if k ≥ 0 and Fk is the free algebra with generators
y1 , . . . , yk , then pk (y1 , . . . , yk ) = CBHk (y1 , . . . , yk ). If now i = (i1 , . . . , ik ) is a sequence of
elements of {1, . . . , n}, we have a unique morphism φi : fk → fn , such that y1 7→ xi1 , . . . ,
yk 7→ xik .
Then
pn (xi1 · · · xik ) = pfn ◦ U (φi )(y1 · · · yk ) = φi ◦ pfk (y1 · · · yk )
= φi (CBHk (y1 , . . . , yk )) = CBHk (xi1 , . . . , xik ) = cbhn (xi1 · · · xik ),
which proves the lemma.
End of proof of Proposition 1. We denote by pbn : Fbn → bfn the map similarly derived from
b k≥0 S k (bfn ) (where ⊕
b is the direct product). Then pn = cbhn implies
the isomorphism Fbn ' ⊕
d
pbn = cbhn .
If now X ∈ Fbn is group-like, let ` := log(X). We have X = 1+`+`2 /2!+· · · ; here `k ∈ S k (bfn ),
d n (X) = ` = log(X).
so pbn (X) = `. Hence cbh
2
Corollaries
The KZ associator is defined as follows. Let A0 , A1 be noncommutative variables. Let F2 be
the free associative algebra generated by A0 and A1 , let f2 ⊂ F2 be its (free) Lie subalgebra
generated by A0 and A1 . Let Fb2 and bf2 be the degree completions of F2 and f2 (A0 and A1 have
degree 1).
The KZ associator Φ is defined [1] as the renormalized holonomy from 0 to 1 of the differential
equation
A0
A1
0
G (z) =
+
G(z),
(1)
z
z−1
A0 as z → 0+
b b
i.e., Φ = G1 G−1
0 , where G0 , G1 ∈ F2 ⊗ O]0,1[ are the solutions of (1) with G0 (z) ∼ z
A
−
and G1 (z) ∼ (1 − z) 1 as z → 1 ; here O]0,1[ is the ring of analytic functions on ]0, 1[, and
b V is the completion of F2 ⊗ V w.r.t. the topology defined by the F2≥n ⊗ V (here F2≥n is
Fb2 ⊗
the part of F2 of degree ≥ n).
We recall Le and Murakami’s formula for Φ [3]. We say that a sequence (a1 , . . . , an ) ∈ {0, 1}n
is admissible if a1 = 1 and an = 0. If (a1 , . . . , an ) is admissible, we set
Z
1
ω a1 ◦ · · · ◦ ω an ,
ωa1 ,...,an =
0
Rb
R
where ω0 (t) = dt/t, ω1 (t) = dt/(t − 1) and a α1 ◦ · · · ◦ αn = a≤t1 ≤···≤tn ≤b α1 (t1 ) ∧ · · · ∧ αn (tn ).
Up to sign, the ωa1 ,...,an are MZV’s (multiple zeta values).
A Formula for the Logarithm of the KZ Associator
3
If (i1 , . . . , in ) is an arbitrary sequence in {0, 1}n , and (a1 , . . . , an ) is an admissible sequence,
,...,an
by the relation
define integers Cia11,...,i
n
X
,...,an
Ain · · · Ai1
Cia11,...,i
n
(i1 ,...,in )∈{0,1}n
card(T )
X
=
(−1)card(S)+card(T ) A1
card(S)
A(a1 , . . . , an )S,T A0
,
S ⊂ {α|aα = 0},
T ⊂ {β|aβ = 1}
where for any S ⊂ {α|aα = 0}, T ⊂ {β|aβ = 1}, A(a1 , . . . , an )S,T :=
Q
Aaα (the
α∈[1,n]\(S∪T )
product is taken in decreasing order of the α’s).
Theorem 1 ([3]).
X
Φ=1+
X
,...,an
Ain · · · Ai1 .
ωa1 ,...,an Cia11,...,i
n
n≥1 (a1 , . . . , an ) admissible
(i1 , . . . , in ) ∈ {0, 1}n
d 2 (Φ), therefore:
Since Φ ∈ Fb2 is a group-like element, Proposition 1 implies that log(Φ) = cbh
Corollary 1.
log(Φ) =
X
X
,...,an
ωa1 ,...,an Cia11,...,i
CBHn (Ain , . . . , Ai1 ).
n
n≥1 (a1 , . . . , an ) admissible
(i1 , . . . , in ) ∈ {0, 1}n
the logarithm
of the analogue Ψ of
Using the explicit formula of [2], one computes similarly
P
bζ /(z − ζ) G(z).
the KZ associator of the equation G0 (z) = A/z +
ζ|ζ n =1
Proposition 1 also implies:
Lemma 2. Let g be a nilpotent Lie algebra, G be the associated Lie group, let a < b ∈ R.
Fix h(z) ∈ C 0 ([a, b], g) and let H be the holonomy from a to b of the differential equation
H 0 (z) = h(z)H(z), where H(z) ∈ C 1 ([a, b], G). Then
XZ
log(H) =
CBHn (h(zn ), . . . , h(z1 ))dz1 · · · dzn .
n≥1 a≤z1 ≤···≤zn ≤b
Acknowledgements
We first established the formula for log(Φ) in Corollary 1 by analytic computations (using
a direct proof of Lemma 2). It was the referee who remarked its formal similarity with the formula
d 2 (Φ).
of Le and Murakami (Theorem 1); this remark can be expressed as the equality log(Φ) = cbh
This led us to try and understand whether this formula followed from the group-likeness of Φ,
which is indeed the case (Proposition 1). C. Reutenauer then pointed out that a part of our
argument is a result in his book.
[1] Drinfeld V., On quasitriangular quasi-Hopf algebras and a group closely connected with Gal(Q̄/Q), Leningrad
Math. J., 1991, V.2, 829–860.
[2] Enriquez B., Quasi-reflection algebras, multiple polylogarithms at roots of 1, and analogues of the group GT,
math.QA/0408035.
[3] Le T.T.Q., Murakami J., Kontsevich’s integral for the Kauffman polynomial, Nagoya Math. J., 1996, V.142,
39–65.
[4] Reutenauer C., Free Lie algebras, London Mathematical Society Monographs. New Series, Vol. 7, Oxford
Science Publications, New York, The Clarendon Press, Oxford University Press, 1993.