Classification of bicovariant differential calculi on quantum groups

Abstract

Suppose thatq is not a root of unity. We classify all bicovariant differential calculi of dimension greater than one on the quantum groupsGL q (N),O q (N) andSp q (N) for which the differentials du of the matrix entriesu generate the left module of first order forms. Our first classification theorem asserts that there are precisely two one-parameter families of such calculi onGL q (N) forN≧3. In the limitq→1 only two of these calculi give the ordinary differential calculus onGL(N). Our second main theorem states that apart from finitely manyq there exist precisely two differential calculi with these properties onO q (N) andSp q (N) forN≧4. This strengthens the corresponding result proved in our previous paper [SS2]. There are four such calculi onO q (3). We introduce two new 4-dimensional bicovariant differential calculi onO q (3).