Differential calculi on quantum vector spaces with Hecke-type relations

Abstract

From a vector spaceV equipped with a Yang-Baxter operatorR one may form the r-symmetric algebraS _R V=TV/〈v ⊗w−R(v ⊗w)〉, which is a quantum vector space in the sense of Manin, and the associated quantum matrix algebraM _R V=T(End(V))/〈f ⊗g−R(f ⊗g)R ^-1〉. In the case whenR satisfies a Hecke-type identityR ²=(1−q)R+q, we construct a differential calculus Ω _R V forS _R V which agrees with that constructed by Pusz, Woronowicz, Wess, and Zumino whenR is essentially theR-matrix of GL _q (n). Elements of Ω _R V may be regarded as differential forms on the quantum vector spaceS _R V. We show that Ω _R V isM _R V-covariant in the sense that there is a coaction Φ_*: Ω _R V →M _R V ⊗ Ω _R V with Φ_*d=(1 ⊗ d)Φ_* extending the natural coaction Φ:S _R V →M _R V ⊗S _R V.