Drinfel’d doubles and Lusztig’s symmetries of two-parameter quantum groups. (English) Zbl 1148.17007
Summary: We find the defining structures of two-parameter quantum groups \(U_{r,s}({\mathfrak g})\) corresponding to the orthogonal and the symplectic Lie algebras, which are realized as Drinfel’d doubles. We further investigate the environment conditions upon which the Lusztig’s symmetries exist between \((U_{r,s}({\mathfrak g}), \langle\, , \,\rangle)\) and its associated object \((U_{s^{-1},r^{-1}}({\mathfrak g}), \langle\, |\,\rangle)\).
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 16W35 | Ring-theoretic aspects of quantum groups (MSC2000) |
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