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David Penneys
Professor of Mathematics at The Ohio State University Office: Room 752 Math Tower Email: penneys (dot) 2 (at) osu (dot) edu |
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Unitary Quantum Symmetries Lite
I am in the process of compiling my notes from 3 topics courses at OSU from 2018 (Quantum Algebra), 2021 (Topological Phases of Matter), and 2023 (Higher Linear Algebra) into a book with Giovanni Ferrer and Kyle Kawagoe.
The book has 3 parts. Each part discusses objects, their endomorphisms, a weaker version of equivalence, some important examples, and applications.
I am in the process of compiling my notes from 3 topics courses at OSU from 2018 (Quantum Algebra), 2021 (Topological Phases of Matter), and 2023 (Higher Linear Algebra) into a book with Giovanni Ferrer and Kyle Kawagoe.
The book has 3 parts. Each part discusses objects, their endomorphisms, a weaker version of equivalence, some important examples, and applications.
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Part I: Hilb: Finite dimensional Hilbert spaces.
- Chapter 1: Hilbert spaces and operators, 38 pages
- Chapter 2: Operator algebras and their modules, 42 pages
- Chapter 3: Morita equivalence, 44 pages
- Chapter 4: The Temperley-Lieb-Jones algebras, 41 pages
- Chapter 5: Basic quantum information, 45 pages [still in progress]
- Chapter 6: Lattice models
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Part II: 2Hilb: Finite semisimple unitary categories.
- Chapter 7: Semisimple categories, 41 pages
- Chapter 8: Unitary categories, [in progress]
- Chapter 9: Fusion categories, [in progress]
- Chapter 10: Unitary fusion categories, [in progress]
- Chapter 11: (Unitary) Braided fusion categories, [in progress]
- Chapter 12: Module categories and algebra objects, [in progress]
- Chapter 13: Topological quantum field theory and topological order, [in progress]
- Part III: 3Hilb: Finite semisimple unitary 2-categories.
- References
Files
Here are notes from my 2023 course on Higher Linear Algebra.
\begin{warning} There are known errors in these notes. \end{warning}
Here are other helpful links.
Here are notes from my 2023 course on Higher Linear Algebra.
- Motivational charts for the course
- Notes on finite dimensional Hilbert spaces and operator algebras
- Notes on (unitary) categories and 2-vector/Hilbert spaces
- Notes in progress on (unitary) tensor categories
- Notes in progress on (unitary) 2-categories
- Notes on finite dimensional operator algebras
- Notes on basic quantum information
- Notes on lattice models
- Notes on fusion categories
- Notes in progress on quantum constructions from unitary fusion categories
- Notes on unitary modular tensor categories
- Synoptic chart of tensor categories [animated video]
- Notes on 2-categories
- Notes on 3-categories
- Notes in progress on Phase transitions
\begin{warning} There are known errors in these notes. \end{warning}
Here are other helpful links.
- Quantum information meets quantum matter, Bei Zeng, Xie Chen, Duan-Lu Zhou, Xiao-Gang Wen
- Topological Quantum Computation, Zhenghan Wang
- Topological Quantum: Lecture Notes and Proto-Book, Steven H. Simon
Announcements for UQSL
2026/7/8, 16:39
Chapters 5 and 7
Drafts of Chapters 5 and 7 are posted. Chapter 5 is still very much a work in progress, but we are happier with Chapter 7. We expect unitary categories to be live soon.
2025/5/15, 16:50
Chapters 3-4
Drafts of Chapters 3-4 posted. Chapter 3 is still very much a work in progress. We may be able to simplify the exposition -- we are in the process of balancing operator algebraic techniques with the (semi)simplicity of the various unitary versions of representation theory.
2025/4/30, 13:02
Up and running
Drafts of Chapters 1-2 posted. More updates coming soon (in geological time).
2026/7/8, 16:39
Chapters 5 and 7
Drafts of Chapters 5 and 7 are posted. Chapter 5 is still very much a work in progress, but we are happier with Chapter 7. We expect unitary categories to be live soon.
2025/5/15, 16:50
Chapters 3-4
Drafts of Chapters 3-4 posted. Chapter 3 is still very much a work in progress. We may be able to simplify the exposition -- we are in the process of balancing operator algebraic techniques with the (semi)simplicity of the various unitary versions of representation theory.
2025/4/30, 13:02
Up and running
Drafts of Chapters 1-2 posted. More updates coming soon (in geological time).
last modified: 2025/5/15