Classification of bicovariant differential calculi on quantum groups of type A, B, C and D

Abstract

Under the assumptions thatq is not a root of unity and that the differentialsdu of the matrix entries span the left module of first order forms, we classify bicovariant differential calculi on quantum groupsA n−1 ,B n ,C n andD n . We prove that apart one dimensional differential calculi and from finitely many values ofq, there are precisely2n such calculi on the quantum groupA n−1 =SL q (n) forn≧3. All these calculi have the dimensionn 2. For the quantum groupsB n ,C n andD n we show that except for finitely manyq there exist precisely twoN 2-dimensional bicovariant calculi forN≧3, whereN=2n+1 forB n andN=2n forC n ,D n . The structure of these calculi is explicitly described and the corresponding ad-invariant right ideals of ker ε are determined. In the limitq→1 two of the 2n calculi forA n−1 and one of the two calculi forB n ,C n andD n contain the ordinary classical differential calculus on the corresponding Lie group as a quotient.