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, 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 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 Drinfeld-Jimbo and Manin. Self-duality of these objects is expressed by an 'R-matrix', 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, geometric, topological and physical aspects of Hopf algebras.
- Essential: undergraduate linear algebra, group theory, ring theory.
- Advantageous: representation theory, Lie algebras.
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.
- (Co)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: invariants of knots and links arising from representations of a quantum group.