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Fox-Neuwirth cells, quantum shuffle algebras, and the homology of braid groups

Banff International Research Station for Mathematical Innovation and Discovery

The notion of a braided vector space \(V\) comes from the Hopf algebra community, and examples abound, from conjugacy classes in groups to braidings coming from Cartan matrices. From this definition, the tensor powers of \(V\) form a family of braid group representations. They also may be assembled into a non-commutative, non-cocommutative braided Hopf algebra called a quantum shuffle algebra. Our main result identifies the homology of the braid groups with these coefficients as the cohomology of this algebra. Using change of rings spectral sequences, we begin to get a handle on these homology groups. If time permits, we will discuss applications to conjectures in arithmetic statistics. This is joint work with TriThang Tran and Jordan Ellenberg.

Mathematics; Algebraic topology; Category theory; homological algebra

video/mp4

Author affiliation: University of Minnesota

2017-01-25

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/59804

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/