Quantum groups and quantum shuffles
Let \(U^+_q\) be the ``upper triangular part'' of the quantized enveloping algebra associated with a symmetrizable Cartan matrix. It is known that the quantum double construction allows to construct \(U_q\) from its Hopf subalgebra \(U^+_q\). So, in this paper, the author restricts attention to \(U^+_q\). It is shown that \(U^+_q\) is isomorphic (as a Hopf algebra) 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 \(\mathbb{Z}^n\). This method gives supersymmetric as well as multiparameter versions of \(U^+_q\) in a uniform way for a suitable choice of the Hopf bimodule. The author gives a classification result about the Hopf algebras which can be obtained in this way, under a reasonable growth condition. It is also shown how the general formalism allows to reconstruct higher rank quantized enveloping algebras from \(U_q\text{sl}(2)\) and a suitable irreducible finite-dimensional representations.
scientific article; zbMATH DE number 1199917