Algebraic Structures on Grothendieck Groups of a Tower of Algebras

123 Pages·2006·0.53 MB·English

Algebraic Structures on Grothendieck Groups of a
Tower of Algebras
HUILAN LI
Adissertation submitted tothe Faculty of GraduateStudies
in partialfulfillmentof therequirements
for thedegree of
Doctor of Philosophy
GraduateProgrammein Mathematics
York University
Toronto, Ontario
April,2007
Algebraic Structures on Grothendieck Groups of a
Tower of Algebras
byHuilan Li
a dissertation submitted to the Faculty of Graduate Studies of York
Universityinpartialfulfillmentoftherequirementsforthedegreeof
DOCTOR OF PHILOSOPHY
(cid:13)c
Permission has been granted to the LIBRARY OF YORK UNIVER-
SITY to lend or sell copies of this dissertation, to the NATIONAL
LIBRARYOFCANADAtomicrofilm thisdissertation andtolendor
sellcopiesofthefilm,andtoUNIVERSITYMICROFILMStopublish
an abstract ofthis dissertation.
The author reserves other publication rights, and neither the disser-
tation nor extensive extracts from it may be printed or otherwise re-
produced without the author’s written permission.
Abstract
For any algebra there are two Grothendieck groups we are interested in.
Oneisassociatedwiththecategoryoffinitelygeneratedleftmodules. Theother
is associated with the category of finitely generated projective left modules. In
this work, we study the algebraic structure on those two Grothendieck groups
in a more general case, that is, the Grothendieck groups of a tower of algebras.
Wedefinethenotionofatowerofalgebras. Westartwithadirectsumoffinitely
dimensional algebras with an external product which is an injective homomor-
phism of algebras and satisfies associativity. The Grothendieck groups of the
tower are the direct sums of the Grothendieck groups of the component alge-
bras. So they are graded. Using the external product we can define the induc-
tionandrestrictionofmodules. Undercertainaxiomswecanusethedefinitions
ofinduction andrestriction todefineamultiplication, acomultiplication, aunit
anda counit on those two Grothendieck groups, respectively.
The purpose of the thesis is to describe what axioms we need on a tower
ofalgebrassothattheGrothendieckgroupsaregradedbialgebrasandthatthey
are dual to each other. For instance, the Grothendieck group of the tower of
symmetric group algebras has a self-dual graded Hopf algebra structure. It is
isomorphic to the algebra of symmetric functions. Moreover, we give some ex-
amples to indicate why these axioms are necessary. We also give auxiliary re-
sults that are helpful to verify the axioms. We conclude with some remarks on
generalized towers of algebras leading to a structure of generalized bialgebras
(in the sense ofLoday) on theirGrothendieck groups.
iv
Acknowledgement
I would like to express my gratitude to Nantel Bergeron for advising me
insomanyways, valuablecommentsandinsights, andendlessencouragement
andsupport throughout my Ph.D. study.
I am grateful to Ian Goulden and Mike Zabrocki for excellent advice, cor-
rections and wealth ofuseful suggestion.
Thank you to my other committee member, Ilijas Farah, outside member,
Jurij W. Darewych, and Dean’s representative, Steven Wang. My thanks are to
the excellent faculty and staff at the Department of Mathematics and Statistics,
York University, especially Amy Wu, Augustine Wong and Primrose Miranda
for their help and assistance during the past five years. I also thank Christophe
Hohlwegfor useful comments.
Last, but not least, I would like to thank my father, Taiming Li, and my
mother, Decui Chen, and those who, directly or indirectly, helped me to finish
mythesis.
v
Contents
Abstract iv
Acknowledgement v
1 Introduction 1
2 The Tower CS and Motivation 5
n≥0 n
L
2.1 Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Matrix Representations . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 G-Modules and the Group Algebraof G . . . . . . . . . . . . . . . 11
2.4 Reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Complete Reducibilityand Maschke’sTheorem . . . . . . . . . . 16
2.6 G-Homomorphisms and Schur’s Lemma . . . . . . . . . . . . . . 17
2.7 Group Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.8 InnerProducts ofCharacters . . . . . . . . . . . . . . . . . . . . . . 22
2.9 Decomposition of the Group Algebraof G . . . . . . . . . . . . . . 24
2.10 Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
vi
2.11 Induction and Restriction ofRepresentations . . . . . . . . . . . . 27
2.12 Mackey’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.13 The RingofSymmetric Functions . . . . . . . . . . . . . . . . . . . 30
2.14 Schur Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.15 The Characteristic Map . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.16 More Explanation ofthe Motivation . . . . . . . . . . . . . . . . . 37
3 Notation and Propositions 39
3.1 Algebra, Coalgebra and Bialgebra . . . . . . . . . . . . . . . . . . 39
3.2 Antipode and HopfAlgebra . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Duality of Algebraand Coalgebra Structure . . . . . . . . . . . . . 43
3.4 Grothendieck Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.5 Induction and Restriction ofModules . . . . . . . . . . . . . . . . 46
4 Grothendieck Groupsof a Tower of Algebras
(PreservingUnities) 47
4.1 Tower ofAlgebras(Preserving unities) . . . . . . . . . . . . . . . . 48
4.2 Induction and Restriction on G (A) . . . . . . . . . . . . . . . . . . 49
0
4.3 Induction and Restriction on K (A) . . . . . . . . . . . . . . . . . . 53
0
4.4 Pairing on K (A)×G (A) . . . . . . . . . . . . . . . . . . . . . . . 56
0 0
4.5 Main Result1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
vii
4.6 Tower Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 67
5 Grothendieck Groupsof a Tower of Algebras
(not Preservingunities) 71
5.1 Tower ofAlgebras(not Preserving Unities) . . . . . . . . . . . . . 71
5.2 Induction and Restriction on G (A) . . . . . . . . . . . . . . . . . . 73
0
5.3 Induction and Restriction on K (A) . . . . . . . . . . . . . . . . . . 75
0
5.4 Pairing on K (A)×G (A) . . . . . . . . . . . . . . . . . . . . . . . 78
0 0
5.5 Main Result2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6 Examples 87
6.1 ExamplesSatisfying AllAxioms . . . . . . . . . . . . . . . . . . . 87
6.2 Examplesnot Satisfying Condition (V) . . . . . . . . . . . . . . . . 100
7 Future Work 109
Bibliography 113
viii
Chapter 1
Introduction
In 1977, L. Geissinger realized Sym (symmetric functions in infinite variables)
as a self-dual graded Hopf algebra [12]. Using the work of Frobenius and
Schur[32],thiscanbeinterpretedastheself-dualGrothendieckHopfalgebraof
CS
the tower of symmetric group algebras . Since then, mathematicians
n≥0 n
L
have encountered many instances of combinatorial Hopf algebras. In several
instances, they study a pair of dual graded Hopf algebras, and find that this
duality can be interpreted as the duality of the Grothendieck groups of an ap-
propriate tower of algebras. For example, C. Malvenuto and C. Reutenauer
established the duality between the Hopf algebra of NSym (noncommutative
symmetric functions) and the Hopf algebra of QSym (quasi-symmetric func-
tions) when looking at the combinatorics of descents [22]. Later, D. Krob and
J. -Y. Thibon showed that this duality can be interpreted as the duality of the
Grothendieck groups associated to H (0), the tower of Hecke algebras at
n≥0 n
L
q = 0 [19]. More recently, N. Bergeron, F. Hivert, and J. -Y. Thibon showed that
if one uses HCl (0), the tower of Hecke-Clifford algebras at q = 0, then
n≥0 n
L
1
one gets a similar interpretation for the duality between the Peak algebra and
its dual [4]. In [29] Sergeev constructed semi-simple super algebras Se (n ≥ 0)
n
and a characteristic map from the super modules of Se= Se to Schur’s Q-
n n
L
functions Γ = C[p ,p ,...] ⊆Sym. From results which will appear in this thesis
1 3
wecouldre-derivetheclassicalresultthatΓisaself-dualgradedHopfsubalge-
bra of Sym.
The relations among all these Hopf algebras and their corresponding towers
canbefoundinthefollowingdiagramandtable. Thealgebrasbelowthedashed
line correspond to the Grothendieck groups in the category of finitely gener-
ated modules and those above the dashed line correspond to the Grothendieck
groups in the category offinitely generated projective modules.
Hopf Algebra Sym QSym Γ Peak
TheDual Algebra Sym NSym Γ Peak∗
Corresponding Tower CS H (0) Se HCl (0)
n n n n n n n n
L L L L
2
In this work, westudy the relationship betweensome graded algebrasAand
thealgebraicstructureontheirGrothendieckgroupsG (A)andK (A)inamore
0 0
general case. More precisely, (A = A ,ρ) is a graded algebra where each
n≥0 n
L
componentA isitselfanalgebraandρisanexternalmultiplication. Wewillcall
n
A a tower of algebras if it satisfies some axioms given in our work. No formal
study of this kind has been done so far. Up to this point it was not clear what
were the right conditions to impose on a tower of algebras to get the desired
algebraicstructure on theirGrothendieck groups. Inthisthesis, wefindalistof
axiomson a tower ofalgebraswhich will implythattheirGrothendieck groups
are graded Hopf algebras. Moreover, we define a pairing and show that the
corresponding Grothendieck groups are dual to each other as Hopf algebras if
the tower ofalgebras satisfies anadditional condition.
This dissertation is divided into six chapters as follows. In Chapter 2 we re-
view some preliminary propositions and explain our motivation. In Chapter 3
werecallsomedefinitionsandpropositions aboutbialgebrasandGrothendieck
groups. In Chapter 4 we discuss the axioms on a tower of algebras (A =
A ,ρ) with ρ preserving unities so that their Grothendieck groups are
n≥0 n
L
graded Hopf algebras. We define a pairing and show that the Grothendieck
groups are dual to each other as Hopf algebras. Moreover, we introduce a def-
inition of tower homomorphisms. In Chapter 5 we weaken the condition on ρ
and modify the definitions of induction and restriction to get results similar to
those in Chapter 4. In Chapter 6 we give some examples to indicate that the
Grothendieck groups of a tower of algebras satisfying these axioms are Hopf
algebras dual to each other, and that these axioms are necessary. For some ex-
3