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Hopf algebra techniques applied to the quantum group \(U_ q(sl(2))\). (English) Zbl 0796.17006
Deformation theory and quantum groups with applications to mathematical physics, Proc. AMS-IMS-SIAM Jt. Summer Res. Conf., Amherst/MA (USA) 1990, Contemp. Math. 134, 309-323 (1992).
[For the entire collection see Zbl 0755.00012.]
The full dual Hopf algebra of \(U_ q(\text{sl}(n))\) is studied. \(U_ q(\text{sl}(n))\) and \(k[\text{SL}_ q(n)]\) are naturally paired. If \(q\) is not a root of 1 and \(\text{char}(k)\neq 2\) then all finite-dimensional representations of \(U_ q(\text{sl}(n))\) are completely reducible, \(U_ q(\text{sl}(n))^ 0\) is cosemisimple and decomposes into a semidirect product \(U_ q(\text{sl} (n))^ 0\cong k[\text{SL}_ q(n)] \rtimes \langle \gamma_ 1,\dots, \gamma_{n-1}\rangle\) where the \(\gamma_ i\) are Hopf algebra automorphisms of order 2 of \(k[\text{SL}_ q(n)]\) commuting with each other. The case of \(\text{char} (k)=2\) resp. \(q\) a root of 1 is studied, too, using the quantum hyperalgebra \(\widetilde{U}_ q (\text{sl} (n))\). In the case of \(n=2\) the structure of the Hopf algebras involved, of the integral and of the irreducible representations is given explicitly. An interesting and illuminating survey of results mainly from the literature showing which of the various notions fit together best.

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
14L17 Affine algebraic groups, hyperalgebra constructions
16S35 Twisted and skew group rings, crossed products
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