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Bicovariant differential calculus on quantum groups \(SU_ q(N)\) and \(SO_ q(N)\). (English) Zbl 0743.17015
Following Woronowicz’s proposal, the bicovariant differential calculus on the quantum groups \(SU_ q(N)\) and \(SO_ q(N)\) is constructed. First of all, the authors construct the fundamental bicovariant bimodules of the quantum groups \(SU_ q(N)\) and \(SO_ q(N)\) by using the \(R_ q\) matrix. Other bicovariant bimodules can be constructed using them as building blocks. They also show that the Hopf algebras generated by the linear functionals which relate left and right multiplication of these bicovariant bimodules coincide with the \(q\)-deformed universal enveloping algebras defined by Drinfel’d and Jimbo earlier. Imposing the conditions of bicovariance and consistency with the quantum group structure, they define the differential algebras and exterior derivatives. As an application, they derive the Maurer-Cartan equations and the \(q\)-analogue of the structure constants.

17B37 Quantum groups (quantized enveloping algebras) and related deformations
46L85 Noncommutative topology
46L87 Noncommutative differential geometry
58A15 Exterior differential systems (Cartan theory)
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
Full Text: DOI
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