On IAn and derivations of free Lie algebras [PhD thesis]

114 Pages·2010·0.628 MB·English

by  Tao Jin

On IA and Derivations of Free
n
Lie Algebras
by
Tao Jin
Submitted in Partial Fulfillment
of the
Requirements for the Degree
Doctor of Philosophy
Supervised by
Professor Frederick R. Cohen
Department of Mathematics
Arts, Sciences and Engineering
School of Arts and Sciences
University of Rochester
Rochester, New York
2010
ii
Curriculum Vitae
The author was born in Wuhan, Hubei Provence, China. Before joining the
Department of Mathematics, University of Rochester (UR) in September 2003,
he held dual Bachelors’ degree in Economics and Mathematics from Wuhan Uni-
versity, and a Master of Science degree in Probability and Statistics from Peking
University, China. At the University of Rochester, the author studied algebraic
topologyandrelatedsubjectsundertheguidanceofProfessorFrederickCohen. In
May 2005, he received his Master of Arts degree in Mathematics from UR. This
Ph.D. dissertation is written by the author under the supervision of Professor
Cohen.
iii
Acknowledgments
The author would like to thank his advisor Professor Frederick Cohen. It
was Professor Cohen who introduced him to the study of this interesting subject.
He deeply appreciates Professor Cohen’s enthusiasm, encouragement and direc-
tion. The author would also like to thank Professor Jonathan Pakianathan for the
helpful instructions and discussions in the past five years. His thanks also go to
Alexandra Pettet and Aaron Heap for discussions in the subject of this thesis, and
to Professors Michael Gage, Naomi Jochnowitz and John Harper, to department
secretary Joan Robinson, to Qiang Sun, Ryan Budney, Ryan Dahl, Justin Suki-
ennik, Micah Milinovich, Rui Hu, Lei Liang, and many other people who helped
the author in various ways.
Finally, the author’s appreciation goes to his beloved wife, Shan, and his par-
ents for their love and support.
iv
Abstract
This thesis gives a new, additional structure for the Lie algebra of derivations
of free Lie algebras.
The new structure arises from actions of a classical algebra known as the Schur
algebra. The thesis gives additional structure for (i) the Schur algebras, (ii) their
action on the Lie algebra of derivations of a free Lie algebra, and (iii) the operad
structures constructed from the Schur algebras and their subalgebras which act
on the Lie algebra of derivations of a free Lie algebra. In addition, the entire Lie
algebra of derivations is shown to be generated by quadratic derivations together
with the action of the Schur operad.
Applications to (1) certain subgroups of the automorphism group of a finitely
generated free group are given as well as to (2) comparisons of two intensively
studied filtrations on these groups.
v
Table of Contents
Curriculum Vitae ii
Acknowledgments iii
Abstract iv
1 Introduction 1
1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Structure of the Manuscript . . . . . . . . . . . . . . . . . . . . . 10
2 Preliminaries 12
2.1 Groups and their Filtrations . . . . . . . . . . . . . . . . . . . . . 12
2.2 Free Lie Algebras and (Lie) Bracketing . . . . . . . . . . . . . . . 16
2.3 Derivations of Free Lie Algebras and the Johnson Homomorphism 25
3 Calculation with the Derivations 32
3.1 Starting with Der (L (X )) . . . . . . . . . . . . . . . . . . . . . 32
∗ K 2
3.2 A Fact about χ ’s . . . . . . . . . . . . . . . . . . . . . . . . . . 41
i,j
3.3 On Analogues of the Andreadakis’ Conjecture . . . . . . . . . . . 49
vi
4 Action of the Schur Algebra and Its Sub-algebras 50
4.1 The Schur Algebra on V (K)⊗q . . . . . . . . . . . . . . . . . . . 50
n
4.2 Action of the Schur Algebra on Der (L (X )) . . . . . . . . . . . 62
∗ K n
4.3 ActionofDiagonal-SymmetricSub-AlgebraDS(n,K)onDer (L (X )) 70
∗ K n
4.4 Action of Diagonal Sub-Algebra D(n,K) on Der (L (X )) . . . . 73
∗ K n
4.5 The Schur algebra Sch(n,K) as a graded algebra . . . . . . . . . 75
4.6 The Schur Operad . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.7 Diagonal Operad . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5 Short Exact Sequences of Lie Algebras 88
5.1 Generalized Kohno-Falk-Randell Theorem . . . . . . . . . . . . . 88
5.2 Short Exact Sequences of Johnson Lie Algebras . . . . . . . . . . 91
6 Some Speculation 96
6.1 An Algorithm to Check Special Cases of the Andreadakis’ Conjecture 96
6.2 A Conjecture on Checking Elements in IA . . . . . . . . . . . . . 100
n
6.3 The subgroup generated by χ , χ and χ . . . . . . . . . . . . 102
i,j j,k k,i
Bibliography 104
1
1 Introduction
1.1 General
Given set X, let F(X) be the free group generated by X. Let X = {x ,...,x },
n 1 n
then F(X ) denotes the free group generated by X . Let Aut(F(X )) be the
n n n
automorphism group of group F(X ). The group IA is defined to be the kernel
n n
of the natural quotient
Aut(F(X )) → GL(n,Z)
n
.
f (cid:55)→ f : H (F(X )) → H (F(X ))
∗ 1 n 1 n
This map is an epimorphism since GL(n,Z) is generated by the elementary ma-
trices which are in the image of this homomorphism. Thus there is the following
short exact sequence of groups
1 → IA → Aut(F(X )) → GL(n,Z) → 1. (1.1)
n n
Although Nielsen studied IA as early as in 1917 [N], the structure of IA
n n
is still not well-understood. In 1997, Krsti´c and McCool proved that IA is not
3
finitely presentable [KMc, Theorem 1], and also showed that H2(IA ) is infinitely
3
generated. In 2005, A. Pettet revealed some properties of the second rational
cohomology of IA [Pe]. In 2007, M. Bestvina, K.-U. Bux and D. Margalit proved
n
2
some properties about a certain subgroup of OutF(X ), the outer automorphism
n
group of F(X ), and its homology [BeBuMar]. Some more information can be
n
found in [Mc; CP2; CPVW].
To study IA and Aut(F(X )), the so-called Johnson filtration was first in-
n n
troduced by S. Andreadakis in 1965 [A]. The Johnson Lie algebra grJ(IA ) is the
∗ n
Lie algebra structure associated with the Johnson filtration of group IA . It is
n
found that group IA is related to Der(cid:52)(L (X )), the Lie algebra of derivations of
n ∗ Z n
the free Lie algebra L (X ), via the so-called Johnson homomorphism, which was
Z n
studied by N. Kawazumi years ago (see, for example, [Ka]). It is known that the
Johnson homomorphism is a monomorphism of Lie algebras from the Johnson Lie
algebragrJ(IA )toDer (L (X )), thesub-LiealgebraofDer(cid:52)(L (X ))consisting
∗ n ∗ Z n ∗ Z n
of all the derivations of degree at least 4 (see, for example, [CP1]).1 For reasons
that will be explained later, Der (L (X )), rather than Der(cid:52)(L (X )), will be
∗ Z n ∗ Z n
the main object of this thesis, and it is still called the Lie algebra of derivations
of the free Lie algebra L (X ).
Z n
The Lie algebra of derivations of a free Lie algebra has been an object of
study in its own right, which appears not only in geometric topology but also
in algebraic topology as well as number theory. For example, the Lie algebra
of derivations of a free Lie algebra is related to the study of the mapping class
group M of closed oriented surface Σ of genus g and its subgroup I , the Torelli
g g g
group. S.Moritagavesubstantialanddeepinformationinhissurveypapers[Mo1;
Mo2]. Recently, A. Heap studied some invariants of the Johnson filtration of the
mapping class group [H].
In 1979, G. V. Belyi proved that the absolute Galois group Gal(Q/Q) can be
embedded in some subgroup of Aut(F(cid:98) ), where F(cid:98) is the pro-finite completion of
2 2
1Here, the degree of a derivation is twice of the original degree. Such a treatment is adopted
in order to avoid the difficulty of signs in graded objects. Details about such a treatment of
doubling the degrees will be given later.
3
F [Bel; I]. It turns out that the automorphism group of a pro-finite fundamental
2
group (a special case is pro-finite free groups) and the Lie algebra of derivations
of a free Lie algebra is related to Gal(Q/Q) and Galois representations, see, for
example, [T]. Some other objects are also related to the derivations of free Lie
algebras, for instance, in 2006, L. Schneps proved a combinatorial formula for the
Poisson bracket of two elements of the free Lie algebra on two generators, which
has some nice properties [Sc].
One of the main goals of this thesis is to introduce a new additional structure
for the Lie algebra of derivations of a free Lie algebra, which fits in the above
contexts and arises from actions of a classical algebra known as the Schur algebra,
which was first studied by I. Schur in the early twentieth century to explore the
representation theory of the symmetric group Σ and the general linear group
r
GL(m,C) [Sch]. This algebra traditionally occurs outside of the subject of the
Lie algebra of derivations of a free Lie algebra.
Throughout this thesis, the symbol N stands for the set of all the natural
numbersstartingfrom0andZ standsforallthepositiveintegers. ThesymbolK
+
stands for a commutative ring with identity, which contains at least two elements
so that the trivial case of K = {0} is excluded. All the modules mentioned in this
thesis are unitary, i.e., for any element a in a left R-module A, where R is a ring
with identity element 1 ,
R
1 a = a.
R
LetV (K)denotethefreeK-modulegeneratedbyX andΣ bethesymmetric
n n q
group on q letters, q ≥ 0. When q = 0, Σ = {1} by convention. It is known that
q
Σ acts on V (K)⊗q from the left in a natural way, i.e.,
q n
σ ·(y ⊗···⊗y ) = y ⊗···⊗y (1.2)
1 q σ−1(1) σ−1(q)
for σ ∈ Σ and a typical element y ⊗···⊗y ∈ V⊗q.2
q 1 q n
2Note that whether this action is from the left or the right is determined by how one defines
4
Informula(1.2),thereisanissueaboutthesignsthattheactionofasymmetric
group may introduce. In order to avoid the difficulty of signs induced by grading,
nonzero elements in graded objects including graded modules and (Lie) algebras
are assigned even degrees. The easiest way to do this is to double the original
degree, and all the graded objects occurred in this thesis will be treated this way.
For instance, the tensor algebra T(V (K)) is usually defined as
n
∞
(cid:77)
T(V (K)) = V (K)⊗q. (1.3)
n n
q=0
In order to regrade the tensor algebra, define
T(cid:101)2q(V (K)) = V (K)⊗q and T(cid:101)2q+1(V (K)) = 0 for q ∈ N (1.4)
n n n
and
∞
(cid:77)
T(cid:101)(V (K)) = T(cid:101)q(V (K)). (1.5)
n n
q=0
It is clear that T(V (K)) and T(cid:101)(V (K)) are essentially the same. However, in
n n
many cases, the use of T(cid:101)(V (K)) and T(cid:101)q(V (K)) makes notations more compli-
n n
cated than that of T(V (K)) and V (K)⊗q does. For this reason, T(V (K)) and
n n n
V (K)⊗q are still used in this thesis, although an element in V (K)⊗q is said to be
n n
of degree 2q.
In the above expressions, the notation “⊗” is adopted instead of “⊗ ” for
K
convenience. In this paper, “⊗” always means the tensor product over K unless
other specified. Again, when q = 0, V (K)⊗q = K, and the action of Σ on
n q
V (K)⊗q is just trivial.
n
the multiplication in the symmetric group Σ . Here the treatment in [Hu] is adopted that the
q
product στ of two elements of Σ is the composition function τ followed by σ; that is, the
q
function on the set {1,...,q} given by k (cid:55)→σ(τ(k)).