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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.

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