Q-Analogue of a Twisted Group Ring

Abstract

As is well known, group rings of Coxeter groups have their q-analogue, which are called Hecke algebras. Its natural generalization seems to be q-deformation of twisted group rings. But few examples are found. Since a twisted group ring is a direct summand of the group ring of its representation group in the sense of projective representations, if the group ring of the representation group has a q-deformation, we can obtain a q-analogue of the twisted group ring by taking a direct summand of it, or equivalently, by taking a quotient of it. For example, if a Coxeter group is a representation group of a certain quotient group of it, we can find an example of a q-analogue of a twisted group ring. (In the above, we consider group rings over a field of characteristic zero.) To summarize, we consider the following situation.