Noncommutative Independence from the Braid Group $${\mathbb{B}_{\infty}}$$

Noncommutative Independence from the Braid Group B ∞

We introduce 'braidability' as a new symmetry for (infinite) sequences of noncommutative random variables related to representations of the braid group B∞. It provides an extension of exchangeability which is tied to the symmetric group S∞. Our key result is that braidability implies spread-ability and thus conditional independence, according to the noncommutative extended de Finetti theorem [K...

The Braid Group of Z

We define quasi-Garside groups and prove a theorem about them parallel to Garside’s result on the word problem for the usual braid groups. We introduce a group which we call the braid group B of Zn, and which bears some resemblance to mapping class groups. We prove that B is a quasi-Garside group. We give a ‘small presentation’ for B.

Braid Group Cryptography

In the last decade, a number of public key cryptosystems based on combinatorial group theoretic problems in braid groups have been proposed. We survey these cryptosystems and some known attacks on them. This survey includes: Basic facts on braid groups and on the Garside normal form of its elements, some known algorithms for solving the word problem in the braid group, the major public-key cryp...

Derivative Varieties and the Pure Braid Group

On a Braid Group Action

We discuss some consequences of the braid group action on a categorified quantum group. Results include a description of reflection functors for quiver Hecke algebras and a theory of restricting categorical representations along a face.