Alternating subalgebras of Hecke algebras and alternating subgroups of   braid groups
  For a Coxeter system (G,S) the multi-parametric alternating subalgebra H^+(G) of the Hecke algebra and the alternating subgroup B^+(G) of the braid group are defined. Two presentations for H^+(G) and B^+(G) are given; one generalizes the Bourbaki presentation for the alternating subgroups of Coxeter groups, another one uses generators related to edges of the Coxeter graph. 
O. V. Ogievetsky, L. Poulain d'Andecy
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