Course details
Semester
 Spring 2025
 Monday, January 27th to Friday, April 4th
Hours
 Live lecture hours
 10
 Recorded lecture hours
 10
 Total advised study hours
 80
Timetable
 Wednesdays
 10:05  10:55 (UK)
Course forum
Visit the https://mathsmagic.ac.uk/forums/magic109introductiontohopfalgebrasandquantumgroups
Description
In the course we adopt a handson approach to Hopf algebras: we build on accessible examples arising from groups and Lie algebras, and learn to present new algebras by generators and relations. We develop tensor calculus and emphasise the idea of duality between algebras and coalgebras, modules and comodules etc to approach the celebrated quantum group constructions due to DrinfeldJimbo and Manin. Selfduality of these objects is expressed by an 'Rmatrix', or quasitriangular structure as popularised by Drinfeld and Majid. To give an example of an application of quantum groups, we look at a quasitriangular structure for the quantum group Uq(sl2) to see how it gives rise to knot invariants.
It should be noted that our approach is purely algebraic; the course aims to equip the students with a suitable background to further explore analytic, numbertheoretic, geometric, topological and physical aspects of Hopf algebras.
Prerequisites
 Essential: undergraduate linear algebra, group theory, ring theory.
 Advantageous: representation theory, Lie algebras.
Syllabus
 Linear and multilinear algebra: tensor products, dual spaces, quotients.
 Presentation of algebras using generators and relations. Symmetric algebras, universal enveloping algebras.
 Coalgebras and their representations. (If time: the fundamental theorem on coalgebras.)
 Bialgebras and Hopf algebras. Sweedler notation. Examples, e.g. group algebras.
 Application: proof of the PoincareBirkhoffWitt theorem using Hopf algebra properties.
 (Co)actions of Hopf algebras on algebras. Quantum symmetries.
 Duality pairing. (If time: the Drinfeld double; the Heisenberg algebra.)
 The DrinfeldJimbo quantum group Uq(sl2). qcalculus and qdeformations.
 Quasitriangular structures. The quantum YangBaxter equation. Braidings. (If time: selfduality and the discrete Fourier transform.)
 Application: invariants of knots and links arising from representations of a quantum group.
Lecturer

Dr Yuri Bazlov
 University
 University of Manchester
Bibliography
No bibliography has been specified for this course.
Assessment
The assessment for this course will be released on Tuesday 22nd April 2025 at 00:00 and is due in before Friday 2nd May 2025 at 11:00.
Assessment for all MAGIC courses is via takehome exam which will be made available at the release date (the start of the exam period).
You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).
If you have kept uptodate with the course, the expectation is it should take at most 3 hoursâ€™ work to attain the pass mark, which is 50%.
Please note that you are not registered for assessment on this course.
Files
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Lectures
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