Quantum Superalgebras at Roots of Unity and Topological Invariants of Three-manifolds

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by  Sacha C. Blumen

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Quantum Superalgebras at Roots of Unity and
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9 Topological Invariants of Three-manifolds

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. Sacha Carl Blumen
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a B.A. (Hons), A.Mus.A.
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7 A thesis submitted in fulfilment of the requirements
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for the degree of Doctor of Philosophy
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School of Mathematics and Statistics
University of Sydney, NSW, 2006, Australia
November 17th, 2005
Statement of Originality
To the best of my knowledge and belief, this thesis is original and my own work except as
acknowledged in the text. This thesis has not been submitted, in whole or in part, for any
other degree at this or any other university.
Sacha Carl Blumen
i
ii Statement of Originality
Acknowledgements
I think that it is almost always true that a Doctoral thesis is the combined result of the
candidate’s workandthesupportofthepeoplearoundthem, andmythesis isnoexception.
Many people have offered a huge amount of support during my studies, and there are a
number whom I particularly wish to thank. Firstly, I would like to thank Troy Roderick,
who has been a constant source of support, especially when I felt that I might never
complete the thesis. He helped me through a lot of times when things were very tough,
and also proffered much fun in all the other parts of life that you can forget about while
writing a thesis. I would also like to thank Br¨onte Metcalf-Roderick.
Thanks to my parents, Colleen and Carl Blumen, for their constant interest in my
studies, from undergraduate days to now.
Thanks to my friends for their support and understanding, especially when it sounded
as if I were singing a continual refrain of “just another 6 months!”.
Thanks to my friends and colleagues in the academic world for just being there and
being a funny sort of ‘family’. In particular, thanks to people in the Department of
Mathematics, University of Queensland, where I started my Ph.D.: in no particular order
(and to just mention a few), thanks to James Wood, Andrew Scott, Maithili Mehta, Nick
Cavenagh, Andrew Blinco, Michiru Takizawa, and Professors Tony Bracken and Mark
Gould. Thanks to the Department of Mathematics for giving me a start in mathematics
and mathematical physics, something I had wanted to do for many years!
Thanks to friends and colleagues in the School of Mathematics and Statistics, the
University of Sydney, where I completed this thesis: fellow Ph.D. students, my office-
mates Mark Hopkins and Annelies Tjetjep for putting up with my tastes in music and the
way I appropriated the office, and everyone else for their friendly faces and willingness to
lend an ear. Thanks to the administrative staff for their help and friendliness, especially
Sonia Morr and Janet Doyle.
Thanks to my Ph.D. supervisor, Dr. Rui Bin Zhang, who suggested the topic of this
thesis and assisted with the multitudes of mathematical problems that appeared, and for
also offering a great deal of insightful criticism of my work. Thanks to the School of
Mathematics and Statistics for providing a space in which I was able to complete this
thesis.
Finally, thanks to everyone else who helped me. Many people gave a lot to help me
during my studies, and I hope that this work stands your support in good stead!
iii
iv Acknowledgements
Abstract
The generalmethod ofReshetikhin and Turaev isfollowed to develop topologicalinvariants
of closed, connected, orientable 3-manifolds from a new class of algebras called pseudo-
modular Hopf algebras. Pseudo-modular Hopf algebras are a class of Z -graded ribbon
2
Hopf algebras that generalise the concept of a modular Hopf algebra.
The quantum superalgebra U (osp(1 2n)) over C is considered with q a primitive Nth
q
|
root of unity for all integers N 3. For such a q, a certain left ideal of U (osp(1 2n)) is
q
≥ I |
(N)
also a two-sided Hopf ideal, and the quotient algebra U (osp(1 2n)) = U (osp(1 2n))/
q q
| | I
is a Z -graded ribbon Hopf algebra.
2
For all n and all N 3, a finite collection of finite dimensional representations of
(N) ≥ (N)
U (osp(1 2n)) is defined. Each such representation of U (osp(1 2n)) is labelled by an
q q
| |
integraldominantweightbelongingtothetruncateddominantWeylchamber. Propertiesof
these representations are considered: the quantum superdimension of each representation
is calculated, each representation is shown to be self-dual, and more importantly, the
decomposition of the tensor product of an arbitrary number of such representations is
obtained for even N.
(N)
It is proved that the quotient algebra U (osp(1 2n)), together with the set of finite
q
|
dimensional representations discussed above, form a pseudo-modular Hopf algebra when
N 6 is twice an odd number.
≥
Using this pseudo-modular Hopf algebra, we construct a topological invariant of 3-
manifolds. This invariant is shown to be different to the topological invariants of 3-
manifolds arising from quantum so(2n+1) at roots of unity.
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vi Abstract
Contents
Statement of Originality i
Acknowledgements iii
Abstract v
1 Introduction 1
1.1 Summary of new results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Notation and background 9
2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Background algebraic structures . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Z -graded quasitriangular Hopf algebras . . . . . . . . . . . . . . . . . . . 15
2
3 Quantum osp(1 2n) at generic q 21
|
3.1 The quantum superalgebra U (osp(1 2n)) . . . . . . . . . . . . . . . . . . . 21
q
|
3.2 Finite dimensional irreducible U (osp(1 2n))-modules . . . . . . . . . . . . 25
q
|
3.3 R-matrices for representations of U (osp(1 2n)) . . . . . . . . . . . . . . . 27
q
|
3.3.1 The universal R-matrix of U (osp(1 2n)) . . . . . . . . . . . . . . . 28
h
|
3.3.2 R-matrices for representations of U (osp(1 2n)) . . . . . . . . . . . 29
q
|
+
3.4 Useful elements of U (g) . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
q
3.5 Spectral decomposition of ˇ . . . . . . . . . . . . . . . . . . . . . . . . 38
V,V
R
3.6 A representation of the Birman-Wenzl-Murakami algebra . . . . . . . . . . 41
3.7 Bratteli diagrams and path algebras . . . . . . . . . . . . . . . . . . . . . . 44
3.7.1 Bratteli diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.7.2 Path algebras related to Bratteli diagrams . . . . . . . . . . . . . . 44
3.7.3 Centraliser algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.8 Projections from V⊗t onto irreducible U (g)-modules . . . . . . . . . . . . 48
q
3.9 Matrix units for . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
t
C
3.9.1 Matrix units in BW ( q2n,q) . . . . . . . . . . . . . . . . . . . . . 53
t
−
3.9.2 Matrix units in . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
t
C
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viii Abstract
4 Quantum osp(1 2n) at roots of unity 69
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4.1 The Z -graded ribbon Hopf algebra U(N)(osp(1 2n)) . . . . . . . . . . . . . 69
2 q
|
(N)
4.1.1 Definition of U (osp(1 2n)) . . . . . . . . . . . . . . . . . . . . . 70
q
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(N)
4.1.2 The universal R-matrix of U (osp(1 2n)) . . . . . . . . . . . . . . 72
q
|
(N)
4.2 U (osp(1 2n))-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
q
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4.2.1 The truncated Weyl alcoves . . . . . . . . . . . . . . . . . . . . . . 74
(N)
4.2.2 U (osp(1 2n))-modules . . . . . . . . . . . . . . . . . . . . . . . . 76
q
|
(N)
4.3 Tensor products of U (osp(1 2n))-modules . . . . . . . . . . . . . . . . . 85
q
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4.3.1 Technical Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3.2 The Tensor Product Theorems . . . . . . . . . . . . . . . . . . . . . 101
4.4 The well-definedness of the projection operators . . . . . . . . . . . . . . . 104
(N)
5 Topological invariants of 3-manifolds from U (osp(1 2n)) 109
q
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5.1 Knots and links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.2 Tangles and ribbon tangles . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.2.1 The category of directed ribbon tangles . . . . . . . . . . . . . . . . 117
5.2.2 Coloured directed ribbon tangles . . . . . . . . . . . . . . . . . . . 120
5.2.3 Representing framed links as (0,0)-ribbon tangles . . . . . . . . . . 123
5.3 The Reshetikhin-Turaev functor F . . . . . . . . . . . . . . . . . . . . . . 124
5.3.1 Calculations using the functor F . . . . . . . . . . . . . . . . . . . 128
5.4 Pseudo-modular Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.4.1 Modular Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.4.2 Pseudo-modular Hopf algebras . . . . . . . . . . . . . . . . . . . . . 136
5.5 Reshetikhin-Turaev invariant arising from pseudo-modular Hopf algebras . 138
(N)
5.6 Invariants of 3-manifolds from U (osp(1 2n)) . . . . . . . . . . . . . . . . 143
q
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5.6.1 Proof of Axioms (I)–(IV) . . . . . . . . . . . . . . . . . . . . . . . . 145
5.6.2 Proof of Axiom (V) . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.6.3 Proof of Axiom (VI) . . . . . . . . . . . . . . . . . . . . . . . . . . 155
(N) (N/2)
5.7 Comparing the invariants from U (osp(1 2n)) and U (so(2n+1)) . . 158
q q
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5.8 Some side results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.8.1 An observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.8.2 A further result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
A Gaussian binomial identities and Gaussian sums 165
A.1 Gaussian binomial identities . . . . . . . . . . . . . . . . . . . . . . . . . . 165
A.2 Gaussian sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
B The q-binomial theorem and generalisations 167
C The Weyl supercharacter formula 173
Abstract ix
D Hopf ideal of U (osp(1 2n)) at roots of unity 175
q
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D.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
D.2 Root vectors in U (osp(1 2n)) . . . . . . . . . . . . . . . . . . . . . . . . . 176
q
|
D.3 The left ideal U (osp(1 2n)) . . . . . . . . . . . . . . . . . . . . . . . . 177
q
I ⊂ |
D.4 The left ideal is a two-sided ideal . . . . . . . . . . . . . . . . . . . . . . 180
D.4.1 (e )N′ aInd (e )N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
i n
D.4.2 Relations between (e )N′, (e )N and e . . . . . . . . . . . . . . . . 182
γ β i
D.4.3 The relations between (e )N′, (e )N and f . . . . . . . . . . . . . . 187
γ β i
D.4.4 The remaining elements of I . . . . . . . . . . . . . . . . . . . . . . 193
D.5 The co-multiplication of e , µ φ . . . . . . . . . . . . . . . . . . . . . . . 193
µ
∈
D.5.1 ∆(e ), 1 i < j n . . . . . . . . . . . . . . . . . . . . . . 194
αi+···+αj ≤ ≤
D.5.2 ∆(e ), i = 1,...,n 1 . . . . . . . . . . . . . . . . . . . . . 195
αi+···+2αn −
D.5.3 ∆(e ), 1 i < j < n . . . . . . . . . . . . . . . . . . 196
αi+···+2αj+···+2αn ≤
D.6 The commutation relations between the components of ∆(e ) . . . . . . . 201
µ
D.6.1 ∆(e ), i = 1,...,n . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
i
D.6.2 ∆(e ), 1 i < j < n . . . . . . . . . . . . . . . . . . . . . . 202
αi+···+αj ≤
D.6.3 ∆(e ), i = 1,...,n 1 . . . . . . . . . . . . . . . . . . . . . 202
αi+···+αn −
D.6.4 ∆(e ), i = 1,...,n 1 . . . . . . . . . . . . . . . . . . . . . 212
αi+···+2αn −
D.6.5 ∆(e ), 1 i < j n 1 . . . . . . . . . . . . . . . . 213
αi+···+2αj+···+2αn ≤ ≤ −
D.7 is a two-sided co-ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
I
D.8 is a two-sided Hopf ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
I
D.9 Technical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
Bibliography 237