Classification of bicovariant differential calculi on quantum groups of type \(A, B, C\) and \(D\)
This paper is devoted to the study of bicovariant differential calculi on quantum groups of the types \(A_{n-1}\), \(B_n\), \(D_n\), and \(C_n\) in the sense of Faddeev, Reshetikhin and Takhtajan, in the case where the deformation parameter \(q\) is not a root of unity. The paper gives a classification of certain bicovariant calculi under the assumption that the differentials of the basic matrix entries generate the module of one forms. The main technical step is to classify certain ad-invariant right ideals in the kernel of the counit, which is done using representation theory. In an introductory section, the authors review the definitions of bicovariant differential calculi and their relation to ad-invariant right ideals of the kernel of the counit. The next part of the paper contains the classification for \(A_{n-1}\) \((n\geq 3)\), the result being that apart from one-dimensional calculi and finitely many values of \(q\), there are precisely \(2n\) calculi satisfying the conditions. The authors also analyze the classical limits of these calculi, showing that precisely two of these \(2n\) calculi have the usual (commutative) calculus on \(\mathrm{SL}(n)\) as a quotient. The second part of the paper treats simultaneously the cases of \(B_n\), \(C_n\) and \(D_n\). In this case, the authors show that except for finitely many values of \(q\), there are precisely two calculi of dimension \(N^2\), where \(N= 2n+1\) for \(B_n\) and \(N=2n\) for \(C_n\) and \(D_n\). In the classical limit, one of these two calculi has the usual calculus on the classical group as a quotient.
scientific article; zbMATH DE number 763765
scientific article