Hecke Algebras of Group Extensions (original) (raw)

Hecke algebras and semigroup crossed product

1999

For an almost normal subgroup Γ0 of a discrete group Γ, conditions are given which allow one to define a universal C∗-norm on the Hecke algebra H(Γ,Γ0). If Γ is a semidirect product of a normal subgroup N containing Γ0 by a group G satisfying some order relations arising from a naturally de-fined subsemigroup T, and if the normalizer of N is also nor-mal in Γ, then a presentation of H(Γ,Γ0) is given. In this situation the C∗-completion of H(Γ,Γ0) is ∗-isomorphic with the semigroup crossed product C∗-algebra C∗(N/Γ0) o T. In their paper introducing a number theoretical model of a quantum sta-tistical system exhibiting a phase transition with symmetry breaking, Bost and Connes introduce the notion of an almost normal subgroup Γ0 of a discrete group Γ, along with the associated Hecke algebra H(Γ,Γ0) and its reduced C∗-algebra completion C∗r (Γ,Γ0) ([BC]). They also provide a pre-sentation of the Hecke algebra in the context of the specific almost normal

Infinitesimal Hecke Algebras II

2009

For W a finite (2-)reflection group and B its (generalized) braid group, we determine the Zariski closure of the image of B inside the corresponding Iwahori-Hecke algebra. The Lie algebra of this closure is reductive and generated in the group algebra of W by the reflections of W. We determine its decomposition in simple factors. In case W is a Coxeter group, we prove that the representations involved are unitarizable when the parameters of the representations have modulus 1 and are close to 1. We consequently determine the topological closure in this case.

Twisted partial actions and extensions of semilattices of groups by groups

International Journal of Algebra and Computation, 2017

We introduce the concept of an extension of a semilattice of groups [Formula: see text] by a group [Formula: see text] and describe all the extensions of this type which are equivalent to the crossed products [Formula: see text] by twisted partial actions [Formula: see text] of [Formula: see text] on [Formula: see text]. As a consequence, we establish a one-to-one correspondence, up to an isomorphism, between twisted partial actions of groups on semilattices of groups and so-called Sieben twisted modules over [Formula: see text]-unitary inverse semigroups.