Infinitesimal Hecke Algebras IV

Abstract

We refine the infinitesimal Hecke algebra associated to a 2-reflection group into a $\mathbb{Z}/2\mathbb{Z}$-graded Lie algebra, as a first step towards a global understanding of a natural $\mathbbm{N}$-graded object. We provide an interpretation of this Lie algebra in terms of the image of the braid group into the Hecke algebra, in connection with unitary structures. This connection is particularly strong when $W$ is a Coxeter group, and when in addition we can use generalizations of Drinfeld's rational even associators. Finally, in the Coxeter ADE case, we provide a generating set of the even part of this Lie algebra, which originates from the rotation subgroup of the Coxeter group.

32 pages

References in corpus (3)

• The freeness conjecture for Hecke algebras of complex reflection groups, and the case of the Hessian group G26
• Automorphisms of complex reflection groups
• Branching properties for the groups G(de,e,r)