## 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

Zbl 0857.16026
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### References:

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