Summary: BRAIDS, Q­BINOMIALS AND QUANTUM GROUPS
MARCELO AGUIAR
Abstract. The classical identities between the q­binomial coefficients and factorials can be gener­
alized to a context where numbers are replaced by braids. More precisely, for every pair i, n of
natural numbers, there is defined an element b (n)
i
of the braid group algebra kBn , 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 recov­
ers the q­identities in this way. These binomial braids b (n)
i
play a crucial role in a simple definition
of a family of quantum groups, including the quantum groups U +
q (C) of Drinfeld and Jimbo.
1. Introduction
The classical identities between the q­binomial coefficients and factorials can be generalized to a
context where numbers are replaced by braids, or more precisely, elements of the braid group algebras
kBn . Thus, for every pair i, n of natural numbers there is defined an element b (n)
i 2 kBn (section 3),
and these satisfy analogs of the classical identities for the binomial coefficients (sections 4 through 8).

Source: Aguiar, Marcelo - Department of Mathematics, Texas A&M University

Collections: Mathematics