# zbMATH — the first resource for mathematics

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)
Full Text: