Abstract
New symmetries (quantum symmetries) formulated in the language of quantum groups are being used more and more in theoretical and mathematical physics. The present paper is devoted to a discussion of bicovariant differential calculus on quantum groups and quantum vector spaces, and is a sequel to a review published in this journal in 1995. A detailed exposition is given of the bicovariant theory of Woronowicz, which is now regarded as the basis for constructing the noncommutative differential geometry on quantum groups. The R -matrix approach is used to give a complete description of the differential calculus on the group GL q (N) and on linear quantum spaces. It is shown how the differential algebra on the group SL q (N) can be obtained (as a subalgebra) from the differential algebra on GL q (N). The problems of the bicovariant differential calculus on the groups SO q (N) and Sp q (2n) are discussed. Essential supplementary information on the general theory of quantum groups, not covered in the first part of the review, is also given.
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