School of Science Department of Mathematics 48 Quantum Groups Supervisor: IP Ivan Chi Ho / MATH Student: SOMER Uras / MATH-IRE Course: UROP 1100, Spring I give an introduction to quantum groups by setting up the prerequisite information. I introduce coalgebras, bialgebras and Hopf algebras. I give examples of quantum groups by deforming familiar matrix algebras, and the universal enveloping algebra of (2). I put emphasis on their representations and coactions after introducing modules and comodules. I give an outline of the FRT construction, which uses the concept of comodules. This main result leads to new solutions for the Yang-Baxter Equation. Quantum Groups Supervisor: IP Ivan Chi Ho / MATH Student: WANGSA Devandhira Wijaya / MATH-PMA Course: UROP 1100, Spring The Compact Quantum Dilogarithm and the Non-Compact Quantum Dilogarithm are fundamental in describing quantum phenomena and have wide-ranging implications in various areas of physics and mathematics. This report aims to validate key identities associated with the Compact Quantum Dilogarithm and the Non-Compact (Faddeev’s) Quantum Dilogarithm; by providing rigorous proofs, we contribute to the understanding and application of these mathematical constructs in the field of quantum theory. Quantum Groups Supervisor: IP Ivan Chi Ho / MATH Student: ZHANG Yitong / MATH-IRE Course: UROP 3100, Fall This is the report for the UROP3100 program, which is considered the continuation of the former UROP2100 and UROP1100 program. In the program, The second chapter of Fuchs’ book Affine Lie algebras and Quantum Groups has been read, and we have started the reading of Lusztig’s paper Tensor structures arising from affine Lie algebra. This report will provide a summary of the reading, following the pathway: 1). Introduction 2). Definitions of affine Lie algebra; 3). Root system and Weyl group of affine Lie algebra; 4). Smooth modules and Sugawara operator; 5). Some further plan.
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