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Prime factor algebras of the coordinate ring of quantum matrices. (English) Zbl 0812.16039

Let \({\mathcal O}_ q(M_ n(k))\) denote the coordinate ring of quantum \(n \times n\)-matrices over a field \(k\) and suppose that \(q\) is not a root of unity. The authors show that every prime ideal of this ring is completely prime. Localizing and factoring they deduce the same result for \({\mathcal O}_ q(GL_ n(k))\) and \({\mathcal O}_ q (SL_ n(k))\).
The proof uses the fact that \({\mathcal O}_ q(M_ n(k))\) is an iterated Ore extension and proceeds by analogy with the proof of the corresponding statement for enveloping algebras of solvable Lie algebras. Thus, the authors obtain a \(q\)-analogue of Sigurdsson’s result. Precisely, they are able to bound the Goldie ranks of prime factors of certain skew polynomial rings \(R[y;\tau, \delta]\). A subtle point is that this latter result is not true without restriction; the authors work with a \((\tau, \delta)\) of the type occurring in quantum groups, namely a \(q\)-skew derivation.

MSC:

16S36 Ordinary and skew polynomial rings and semigroup rings
16D25 Ideals in associative algebras
16P40 Noetherian rings and modules (associative rings and algebras)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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