Iwahori–Hecke algebra

In mathematics, the Iwahori–Hecke algebra, or Hecke algebra, named for Erich Hecke and Nagayoshi Iwahori, is a one-parameter deformation of the group algebra of a Coxeter group.

Hecke algebras are quotients of the group rings of Artin braid groups. This connection found a spectacular application in Vaughan Jones' construction of new invariants of knots. Representations of Hecke algebras led to discovery of quantum groups by Michio Jimbo. Michael Freedman proposed Hecke algebras as a foundation for topological quantum computation.

Start with the following data:

The multiparameter Hecke algebra H_R(W,S,q) is a unital, associative R-algebra with generators T_s for all s ∈ S and relations:

Warning: in recent books and papers, Lusztig has been using a modified form of the quadratic relation that reads ${\displaystyle (T_{s}-q_{s}^{1/2})(T_{s}+q_{s}^{-1/2})=0.}$ After extending the scalars to include the half integer powers q±½
s the resulting Hecke algebra is isomorphic to the previously defined one (but the T_s here corresponds to q½
s T_s in our notation). While this does not change the general theory, many formulae look different.

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