, Volume 23, Issue 2, pp 133-141

Differential calculi on quantum vector spaces with Hecke-type relations

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From a vector spaceV equipped with a Yang-Baxter operatorR one may form the r-symmetric algebraS R V=TV/〈vwR(vw)〉, which is a quantum vector space in the sense of Manin, and the associated quantum matrix algebraM R V=T(End(V))/〈fgR(fg)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 VM R V ⊗ Ω R V with Φ*d=(1 ⊗ d)Φ* extending the natural coaction Φ:S R VM R VS R V.