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Hopf structures on the Borel subalgebra of \(sl(2)\). (English) Zbl 0879.16027
Bureš, J. (ed.) et al., Proceedings of the winter school on geometry and physics, Zdíkov, Czech Republic, January 1993. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 37, 185-199 (1994).
For the Hopf algebra \(\text{Fun}(B)=k[t,t^{-1},p]\), \(\Delta t=t\otimes t\), \(\Delta p=p\otimes t+1\otimes p\) of functions on the Borel subgroup \(B\subset\text{SL}_2\) a class of deformations is considered with commutativity replaced by the more general relation \(tp=\sum_ip^ia_i(t)\). It is shown that the corresponding moduli space is a line with the origin \(\beta=0\) excluded, and with two points at \(\beta=1\). Properties of the “nonstandard” classical limit \(\mathcal T\) at \(\beta=1\) are studied. It is proved that \(\mathcal T\) is selfdual and triangular, and that \(\mathcal T\) is a result of twisting a classical algebra. Correspondingly the whole Hopf algebra \(U^Jsl(2)\) is described. The author cites some other works where the Hopf algebras \(\mathcal T\), \(U^Jsl(2)\) and related ones appear. (The corresponding \(R\)-matrix appeared in the work of Lyubashenko, 1986.) A differential operator realization of \(\mathcal T\) on the Jordanian quantum plane is given.
For the entire collection see [Zbl 0823.00015].

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16S80 Deformations of associative rings
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory