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Finite-dimensional simple modules over certain iterated skew polynomial rings. (English) Zbl 0829.16017
The author constructs and classifies the finite dimensional simple modules over a class of algebras that includes the enveloping algebra \(U(\text{sl}_2(k))\), its quantized analog \(U_q(\text{sl}_2(k))\), and the quantized Weyl algebra \(A_1(k,q)\), as well as other quantum algebras of interest. The algebras in question are of the form \(A[y;\alpha][x;\alpha^{-1},\delta]\) where \(A\) is a commutative affine algebra over an algebraically closed field \(k\) and \(\delta\) is an \(\alpha^{-1}\)-derivation of \(A[y;\alpha]\) such that \(\delta|_A=0\) and \(\delta(y)=u-\rho\alpha(u)\) for some \(u\in A\) and \(\rho\in k^\times\). This construction generalizes the earlier one [introduced by the author in J. Algebra 156, 194-218 (1993; Zbl 0809.16052)], corresponding to the case \(\rho=1\). The prime and primitive ideals of these algebras are investigated by the author [in Math. Proc. Camb. Philos. Soc. 114, 407-425 (1993; Zbl 0804.16028)]. For a further extension and iteration of the construction see the author’s paper [J. Algebra 174, 267-281 (1995)].

MSC:
16S36 Ordinary and skew polynomial rings and semigroup rings
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B35 Universal enveloping (super)algebras
16S30 Universal enveloping algebras of Lie algebras
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