UROP Proceedings 2020-21

School of Science Department of Mathematics 56 Quantum Groups Supervisor: IP Ivan Chi Ho / MATH Student: CHUNG Soobeom / PHYS Course: UROP1100, Spring The quantum groups is associated with certain Hopf algebras which are q-deformation of the enveloping Hopf algebras of semisimple Lie algebras. It is useful in areas of mathematics and physics, such as the quantum inverse scattering and statistical mechanics according to Faddeev (1995) and Polyakov (1984). Here, Hopf algebras’ basic structure and Hopf algebras SLq (2) and Uq(sl(2)) will be introduced, which are related to the classical group SL(2). These are some of the simplest quantum groups and their finite dimensional representations will also be covered. Lastly, the braided bi-algebras, which induce a solution of the YangBaxter equation and guarantee its existence, will be introduced as they can be used to obtain quantum groups GLq(2) and SLq(2). Quantum Groups Supervisor: IP Ivan Chi Ho / MATH Student: CHAN Hong Ming / COSC Course: UROP1100, Summer This report will introduce quantum group by constructing it from the definition of algebra, coalgebra, bialgebra and Hopf algebra. In particular, we are interested in the Hopf algebra Uq(sl(2)) and its representation. Most of the material studied are from Kassel’s book. Some more information on representation theory can be found in Humphrey’s book and we have studied through it before going to [2]. Also an example of using quantum Clebsuh-Gordan formula to find the basis of a 20 dimensional representation of Uq(sl(2)) on Vn ⊗ Vm with n = 4 and m = 3 is given. [2] Christian Kassel. Quantum groups. Springer-Verlag New York, Inc, 1995. Quantum Groups Supervisor: IP Ivan Chi Ho / MATH Student: PACHECO RODRIGUEZ Wemp Santiago / MATH Course: UROP2100, Summer In this report we will study a possible direction for the extension of the classical quantum link invariants introduced by Turaev in 1988 to the setting the modular double . For this sake we will study the ”positive representations” of the modular double and the corresponding universal R-operator as defined by the work of Faddeev, Kashaev and Bytsko-Teschner.

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