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Quantum groups and quantum shuffles. (Groupes quantiques et algèbres de battage quantiques.) (French) Zbl 0929.17005

Let \(U^+_q\) be the ‘upper triangular’ Hopf subalgebra of the quantized enveloping algebra associated with a symmetrizable generalized Cartan matrix. We show that \(U^+_q\) is isomorphic to the subalgebra generated by elements of degree 0 and 1 of the cotensor Hopf algebra associated with a suitable Hopf bimodule on the group algebra of \({Z}^n\). We give a classification result concerning the Hopf algebras which can be obtained by this construction and which satisfy a reasonable growth condition. See the complete version in Invent. Math. 133, 399-416 (1998; Zbl 0912.17005).

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W35 Ring-theoretic aspects of quantum groups (MSC2000)

Citations:

Zbl 0912.17005