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Prime quotients of \(O_ q({\mathfrak m}_ n(k))\). (Quotients premiers de \(O_ q({\mathfrak m}_ n(k))\).) (French) Zbl 0849.16028
Let \(k\) be a field and let \(R\) denote either \({\mathcal O}_q(M_n(k))\), the one-parameter quantized coordinate ring of \(n\times n\) matrices over \(k\), or \(A^{\overline{q},\Gamma}_n(k)\), the multiparameter quantized Weyl algebra of degree \(n\) over \(k\). In the first case, assume that the scalar \(q\in k^\times\) is not a root of unity; in the second, assume that the multiplicative subgroup of \(k^\times\) generated by the entries of the vector \(\overline{q}\in(k^\times)^n\) together with the entries of the matrix \(\Gamma\in M_n(k^\times)\) is torsionfree. It follows from a result of E. S. Letzter and the reviewer that all prime factor rings of \(R\) are integral domains [Proc. Am. Math. Soc. 121, No. 4, 1017-1025 (1994; Zbl 0812.16039)]. Here the author proves that the quotient division ring of any prime factor ring \(R/P\) has the form \(\text{Fract }{\mathcal O}_{\mathbf q}(K^m)\), where \({\mathcal O}_{\mathbf q}(K^m)\) is the multiparameter quantized coordinate ring of affine \(m\)-space over a (commutative) field extension \(K\) of \(k\). (The case \(m=0\) is allowed.) That \(\text{Fract }{\mathcal O}_q(M_n(k))\) has this form had been shown by G. Cliff [J. Lond. Math. Soc., II. Ser. 51, No. 3, 503-513 (1995; Zbl 0835.16013)]. That \(\text{Fract }A^{\overline{q},\Gamma}_n(k)\) has this form follows from work of J. Alev and F. Dumas [J. Algebra 170, No. 1, 229-265 (1994; Zbl 0820.17015)] and D. A. Jordan [J. Algebra 174, No. 1, 267-281 (1995; Zbl 0833.16025)].
Several authors have proved analogous results for the quotient division ring of \(U_q({\mathfrak g})^+\), the positive part of the quantized enveloping algebra of a semisimple Lie algebra \(\mathfrak g\). See J. Alev and F. Dumas [op. cit.] K. Iohara and F. Malikov [Commun. Math. Phys. 164, No. 2, 217-237 (1994; Zbl 0826.17011)] and A. Joseph [C. R. Acad. Sci., Paris, Sér. I 320, No. 12, 1441-1444 (1995; Zbl 0847.17011)].

MSC:
16S36 Ordinary and skew polynomial rings and semigroup rings
16U20 Ore rings, multiplicative sets, Ore localization
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16K40 Infinite-dimensional and general division rings
16U10 Integral domains (associative rings and algebras)
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