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Categorification of \(\mathfrak{sl}_2\) and braid groups. (English) Zbl 1162.20301

de la Peña, José A. (ed.) et al., Trends in representation theory of algebras and related topics. Workshop on representations of algebras and related topics, Querétaro, México, August 11–14, 2004. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3818-0/pbk). Contemporary Mathematics 406, 137-167 (2006).
Summary: We describe categorifications of \(\mathfrak{sl}_2\) and braid groups. In the first part, we give a survey of the case of \(\mathfrak{sl}_2\) (see our joint work with J. Chuang [Ann. Math. (2) 167, No. 1, 245-298 (2008; Zbl 1144.20001)]) and explain how it leads to the construction of derived equivalences. The second part points out the existence of ‘higher symmetries’ in the examples of braid group actions on triangulated categories.
For the entire collection see [Zbl 1092.16500].

MSC:

20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20C08 Hecke algebras and their representations
18E30 Derived categories, triangulated categories (MSC2010)
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
20F36 Braid groups; Artin groups

Citations:

Zbl 1144.20001
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