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Prime spectrum of \(O_q(M_n(k))\): canonical image and normal separation. (Spectre premier de \(O_q(M_n(k))\): image canonique et séparation normale.) (French) Zbl 1024.16001

Let \(k\) be a field with a nonzero element \(q\) which is not a root of one and \(R={\mathcal O}_q(M_n(k))\) the coordinate algebra of quantum \(n\times n\)-matrices over \(k\). The algebra \(R\) is generated by the standard set of variables \(X_{ij}\), \(1\leq i,j\leq n\). Denote by \(\overline R\) the algebra of regular functions on quantum affine space of dimension \(n^2\). It is generated by the same set of generators \(X_{ij}\), \(1\leq i,j\leq n\), but the defining set of relations has the form \(X_{ij}X_{rs}=q^{-1}X_{it}X_{ij}\) if either \(i=r\), \(s<j\) or \(i>r\), \(j=s\) and also the form \(X_{ij}X_{rs}=X_{rs}X_{ij}\) if \(i\neq r\), \(j\neq s\).
It is shown that there exists a canonical injection \(\phi\colon\text{Spec}\;R\to\text{Spec}\;\overline R\) and \(\phi(\text{Spec }R)=\coprod_{w\in W}\text{Spec}_w\overline R\) in the sense of K. A. Brown and K. R. Goodearl [Trans. Am. Math. Soc. 348, No. 6, 2465–2502 (1996; Zbl 0857.16026)]. Moreover, the partition of \(\text{Spec }R\) into subsets \(\phi^{-1}(\text{Spec}_w\overline R)\) is an \(H\)-stratification, where \(H\) is the group of automorphisms of \(R\) under which every element \(X_{ij}\) is an eigenvector. If \(I,P\) are \(H\)-prime ideals and \(I>P\) then there exists \(T\in I\setminus P\) such that \(T\) is \(H\)-normal modulo \(P\). The algebra \(R\) is catenary and if \(R\in\text{Spec }R\) then \(\text{ht}(P)+\text{GKdim}(R/P)=n^2\). There are some other applications of the preceding results.

MSC:

16D25 Ideals in associative algebras
16T20 Ring-theoretic aspects of quantum groups
16S36 Ordinary and skew polynomial rings and semigroup rings
20G42 Quantum groups (quantized function algebras) and their representations

Citations:

Zbl 0857.16026
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References:

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