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Center and Trace of the Twisted Heisenberg Category

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Author(s): Can Ozan Oguz
Series: PhD thesis at University of Southern California
Year: 2018

Language: English

Dedication
ii
Acknowledgements
iii
List Of Tables
vii
List Of Figures
viii
Abstract
ix
Chapter 1: Introduction
1
Chapter 2: Categorification and Decategorification
8
2.1
Categorification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.2
Diagrammatic Categories
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.3
Trace Decategorification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
Chapter 3: Heisenberg Algebras and Heisenberg Categories
23
3.1
Heisenberg Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3.2
Heisenberg Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.2.1
Endomorphism spaces of H . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
3.2.2
H as Categorified Induction and Restriction . . . . . . . . . . . . . . . . . .
28
3.3
Twisted Heisenberg Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3.4
Twisted Heisenberg Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
3.4.1
Endomorphism Spaces of Htw . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.4.2
Htw as Categorified Induction and Restriction
. . . . . . . . . . . . . . . .
42
Chapter 4: Trace of the twisted Heisenberg Category
45
4.1
W-algebras W1+∞ and W − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
4.1.1
Generators of the algebra W −
. . . . . . . . . . . . . . . . . . . . . . . . .
49
4.1.2
Fock space representation of W − . . . . . . . . . . . . . . . . . . . . . . . .
52
4.2
Diagrammatic computations in Htw and in Tr(Htw)
. . . . . . . . . . . . . . . . .
53
4.2.1
Even part of Tr(Htw)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
4.2.2
Grading on Tr(Htw)0
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
4.2.3
Counter-clockwise bubble slide moves
. . . . . . . . . . . . . . . . . . . . .
60
4.2.4
Clockwise bubble slide moves . . . . . . . . . . . . . . . . . . . . . . . . . .
64
4.2.5
Dot degree zero part of Tr(Htw)0 . . . . . . . . . . . . . . . . . . . . . . . .
68
4.2.6
Nonzero dot degree part of Tr(Htw)0 . . . . . . . . . . . . . . . . . . . . . .
71
4.3
Algebra isomorphism between Tr(Htw)0 and W − . . . . . . . . . . . . . . . . . . .
81
4.3.1
Trace of Htw as a vector space
. . . . . . . . . . . . . . . . . . . . . . . . .
82
4.3.2
Generators of the algebra Tr(Htw)0 . . . . . . . . . . . . . . . . . . . . . . .
83
4.3.3
The isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
Chapter 5: Center of the Twisted Heisenberg Category
86
5.1
Sergeev algebra and Type B Weyl group . . . . . . . . . . . . . . . . . . . . . . . .
87
5.1.1
The super representation theory of Sn and �
Bn
. . . . . . . . . . . . . . . .
92
5.1.2
The centers of Sn and C[ �
Bn]
. . . . . . . . . . . . . . . . . . . . . . . . . .
94
5.2
The Subalgebra Γ of Symmetric Functions . . . . . . . . . . . . . . . . . . . . . . .
98
5.3
A new basis of EndHtw(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.4
An isomorphism between EndHtw(1) and Γ . . . . . . . . . . . . . . . . . . . . . . 106
5.5
An action of Tr(Htw)0 on Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.5.1
Description of the action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Reference List
117