Double affine braid group
Encyclopedia
In mathematics, a double affine braid group is a group
containing the braid group
of an affine Weyl group. Their group rings have quotients called double affine Hecke algebra
s in the same way that the group rings of affine braid group
s have quotients that are affine Hecke algebra
s.
For affine An groups, the double affine braid group is the fundamental group
of the space of n distinct points on a 2-dimensional torus.
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
containing the braid group
Braid group
In mathematics, the braid group on n strands, denoted by Bn, is a group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group Sn. Here, n is a natural number; if n > 1, then Bn is an infinite group...
of an affine Weyl group. Their group rings have quotients called double affine Hecke algebra
Double affine Hecke algebra
In mathematics, a double affine Hecke algebra, or Cherednik algebra, is an algebra containing the Hecke algebra of an affine Weyl group, given as the quotient of the group ring of a double affine braid group. They were introduced by Cherednik, who used them to prove Macdonald's constant term...
s in the same way that the group rings of affine braid group
Affine braid group
In mathematics, an affine braid group is a braid group associated to an affine Coxeter system. Their group rings have quotients called affine Hecke algebras. They are subgroups of double affine braid groups.-References:...
s have quotients that are affine Hecke algebra
Affine Hecke algebra
In mathematics, an affine Hecke algebra is the Hecke algebra of an affine Weyl group, and can be used to prove Macdonald's constant term conjecture for Macdonald polynomials.-Definition:...
s.
For affine An groups, the double affine braid group is the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
of the space of n distinct points on a 2-dimensional torus.