Hecke algebras for protonormal subgroups

Abstract

We introduce the term protonormal to refer to a subgroup

Twisted partial actions of Hopf algebras

“…The algebraic version of the latter fact was established in [14]. The importance of partial actions and partial representations was reinforced by R. Exel in [24] where, among other results, it was proved that given a field K of characteristic 0, a group G and subgroups H, N ⊆ G with N normal in G and H normal in N, there is a twisted partial action θ of G/N on the group algebra K(N/H) such that the Hecke algebra H(G, H) is isomorphic to the crossed product K(N/H) * θ G/N. More recent algebraic results on twisted partial actions and corresponding crossed products were obtained in [5], [15] and [30].…”

Recent developments around partial actions

“…Since then the algebraic approach is being developed in diverse directions in various levels of generality, including partial actions of Hopf (or, more generally, weak Hopf) algebras [15,[17][18][19][20][21][22]48,69,72,[78][79][80][81][82][83][86][87][88]165,250,282], semigroups [68,97,132,[197][198][199]209,213,220,227,233,234,242,255], inductive constellations [198], groupoids [37,[40][41][42][43]178], and, more generally, categories [244]. In particular, further algebraic applications have been found to graded algebras [117,121], to Hecke algebras [153], to Leavitt path algebras [184,…”

“…The algebraic version of the latter fact was established in [121]. This algebraic concept was applied to Hecke algebras in [153], where, among other results, it was proved that given a field κ of characteristic 0, a group G and subgroups H, N ⊆ G with N normal in G and H normal in N , there is a twisted partial action θ of G/N on the group algebra κ(N /H ) such that the Hecke algebra H(G, H ) is isomorphic to the crossed product κ(N /H ) * θ G/N . The globalization problem for twisted partial group actions was investigated in [122], whereas other algebraic results on twisted partial actions on rings and corresponding crossed products were obtained in [39,44,49,50,253].…”

Crossed products by twisted partial actions and graded algebras

“…In [27] a machinery was devel-✩ This work was partially supported by CNPq of Brazil and Secretaría de Estado de Universidades e Investigación del MEC, España. oped, based on the interaction between partial actions and partial representations, enabling to study representations and the ideal structure of partial crossed products, and including into consideration prominent examples such as the Toeplitz C * -algebras of quasi-lattice ordered groups, as well as the Cuntz-Krieger algebras. The technique was also used in [26] to define and study Cuntz-Krieger algebras for arbitrary infinite matrices, and recently these ideas were also applied to Hecke algebras defined by subnormal subgroups of length two [25], as well as to C * -algebras generated by crystals and quasi-crystals [6].…”

“…u(m r) = (um )r, u(r 1 r 2 ) = (ur 1 )r 2 , (r 1 r 2 )u = r 1 (r 2 u), (r m )u = r (m u), whereas the property (rψ)r = r(ψ r ) means (ru)r = r(ur ). Moreover,(25) and(26)give u(r m ) = (ur )m and (m r)u = m (ru). It is a straightforward exercise to verify the remaining five equalities, and bellow we check one of them: u(m m) = (um )m. Write m = r i m i in view of M = R M .…”