MAGIC109: Introduction to Hopf algebras and quantum groups

The course will be given in the form of weekly pre-recorded lectures and weekly interactive (tutorial-style) meetings. Students will be expected to watch a pre-recorded lecture and attempt questions on an exercise sheet before each tutorial.

Quantum groups are a manifestation of symmetry in its most abstract algebraic form. One way to motivate quantum groups is to observe that a group acting on a manifold M also acts on its algebra of functions, F(M). One can then substitute F(M) with a more general, noncommutative algebra, which leads to the notion of a Hopf algebra as something capable of acting on such a `noncommutative space'. The theory was reinvigorated by examples due to Drinfeld and Jimbo in the 1980s, inspired by quantum mechanics; the term `quantum group' was coined. Hopf algebras and their representations found applications in many fields including topology, number theory, mathematical physics, quantum information theory and, recently, finance (via stochastic calculus).
In the course we adopt a hands-on approach to Hopf algebras: we build on 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 to approach the celebrated quantum group constructions due to Drinfeld-Jimbo and Manin. Self-duality of these objects is expressed by an 'R-matrix', or quasitriangular structure as popularised by Drinfeld and Majid.

Towards the end of the course, we will work through an application of quantum groups to topology, more specifically knot theory: we will develop an algorithm for calculating the Jones polynomial of a knot from a diagram of the knot. The Jones polynomial is a famous knot invariant which can, for example, distinguish between a trefoil knot and its mirror image. The algorithm will be based on the quasitriangular structure for the quantum group Uq(sl2). This construction is an example of what's known in mathematical physics as a topological quantum field theory, or TQFT.

• Essential: undergraduate linear algebra, group theory, ring theory, undergraduate real analysis (limits and power series).
• Advantageous: representation theory, Lie algebras. 

The course uses language, and rigour, of abstract algebra, All the necessary definitions will be introduced in the course, but students should be prepared to adapt to new notation and new ways of looking at familiar algebraic concepts. 

Topics marked "if time" may be left as optional reading assignments; in this case, they will not be examinable.

• 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 Poincare-Birkhoff-Witt theorem using Hopf algebra properties. 
• Actions of Hopf algebras on algebras. Quantum symmetries. 
• Duality pairing. (If time: the Drinfeld double; the Heisenberg algebra.)
• The Drinfeld-Jimbo quantum group Uq(sl2). q-calculus and q-deformations. 
• Quasitriangular structures. The quantum Yang-Baxter equation. Braidings. (If time: self-duality and the discrete Fourier transform.)
• Application: the Jones polynomial as a quantum knot invariant arising from a representation of Uq(sl2).