Low Dimensional Topology

August 19, 2015

Dongseok Kim 1968-2015

Filed under: Knot theory — dmoskovich @ 11:22 pm

I’m sad to announce the untimely passing of Dongseok Kim, a specialist on Kuperberg spiders and their generalizations. He was also a really nice guy whose conference talks were always well-worth listening to.

Although he’s better known for his quantum $\mathfrak{sl}_3$ and $\mathfrak{sl}_4$ stuff, the part of Kim’s work which was most intriguing for me personally was his work on spiders for Lie superalgebras. His paper on the topic doesn’t seem to be cited, despite superalgebra-related quantum invariants being a hot topic (work of Geer, Patureau-Mirand, Costantino, Turaev…)- has anyone noticed it?

Reshetikhin-Turaev (RT) invariants are constructed in the context of semi-simple categories of representations of quantum groups with some extra structure and satisfying some extra conditions (modular categories). For some quantum groups these categories have nice combinatorial descriptions via Kuperberg’s spiders; the most famous is the Temperley-Leib algebra description of a modular category of $\mathfrak{sl}_2$ representations. For superalgebras the categories are no longer semi-simple, although this probably isn’t a major problem. Kim identifies the following fairly manageable-looking description for a category of quantum $\mathfrak{sl}(1|1)$ representations: