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Quantized enveloping algebras associated with simple Lie superalgebras and their universal \(R\)-matrices. (English) Zbl 0821.17005
For any simple Lie superalgebra \(\mathfrak g\) a quantized superalgebra \(U_{\hbar}({\mathfrak g})\) is explicitly defined in terms of generators and relations. It is proved that \(U_{\hbar}({\mathfrak g})\) is a topologically free superalgebra. The relations include the known \((q)\)-Serre relations and new ones of higher order. The new relations were not classified before even in the classical case.
An analog of the Poincaré-Birkhoff-Witt theorem is proved. The twisted product \(U_{\hbar}({\mathfrak g})^ b\) of \(U_{\hbar}({\mathfrak g})\) with the group algebra of \(Z_ 2\) is introduced. \(U_{\hbar}({\mathfrak g})^ b\) is a usual Hopf algebra which originates from the quantum double of the corresponding Borel subalgebra. The universal \(R\)-matrix for \(U_{\hbar}({\mathfrak g})^ b\) is written down explicitly. It corresponds by restriction to the universal \(R\)-matrix for \(U_ q({\mathfrak g})\) by Khoroshkin-Tolstoy.

17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
17A70 Superalgebras
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