Braids,q-Binomials, and Quantum Groups

Abstract

The classical identities between theq-binomial coefficients and factorials can be generalized to a context in which numbers are replaced by braids. More precisely, for every pairi,nof natural numbers, there is defined an elementb(n)iof the braid group algebrakBn, and these satisfy analogs of the classical identities for the binomial coefficients. By choosing representations of the braid groups, one obtains numerical or matrix realizations of these identities; in particular, one recovers theq-identities in this way. These binomial braidsb(n)iplay a crucial role in a simple definition of a family of quantum groups, including the quantum groupsU+q(C) of Drinfeld and Jimbo. © 1998 Academic Press.