| Abstract: |
| In the classical setting, it is known that the necklace Lie algebra and a graded quotient of the Goldman Lie algebra are isomorphic. This is an example of intersection between 2-dimensional topology and representation theory. There are quantizations of the necklace Lie algebra and the Goldman Lie algebra, called the quantum necklace Lie algebra and the HOMFLYPT skein algebra respectively. However, we did not know any relation between them as the classical setting. In this talk, we define an appropriate filtration on the HOMFLYPT skein algebra and construct an isomorphism between the quantum necklace Lie algebra and the graded quotient of the HOMFLYPT skein algebra. This gives us interactive applications, e.g., a topological interpretation of the quantum necklace Lie algebra and bases of the graded quotient of the HOMFLYPT skein algebra. This is a joint work with Shunsuke Tsuji (Meiji University). If time permits, I will explain the formality problem of the HOMFLYPT skein algebra related to the Kashiwara--Vergne problem. |
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