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Quantum determinantal ideals. (English) Zbl 0958.16025
The quantized coordinate ring $${\mathcal O}_q(M_n(k))$$ of $$n\times n$$ matrices over the field $$k$$ is an affine $$k$$-algebra whose relations reduce, when the parameter $$q$$ is $$1$$, to ones which define the commutative polynomial $$k$$-algebra in $$n^2$$ indeterminates – that is, the classical coordinate ring of $$n\times n$$ matrices over $$k$$. $${\mathcal O}_q(M_n(k))$$ is a bialgebra with the same coproduct as in the classical case, and contains a non-zero central element $$D$$, the quantum determinant, such that $${\mathcal O}_q(M_n(k))/\langle D-1\rangle={\mathcal O}_q(\text{SL}_n(k))$$ and $${\mathcal O}_q(M_n(k))[D^{-1}]={\mathcal O}_q(\text{GL}_n(k))$$ are Hopf algebras, respectively quantum $$\text{SL}_n(k)$$ and quantum $$\text{GL}_n(k)$$. The prime and primitive spectra of these latter two Hopf algebras have been intensively studied – see [A. Joseph, Quantum groups and their primitive ideals, Ergeb. Math. Grenzgeb. (3) 29, Berlin, Springer (1995; Zbl 0808.17004)] and the references there. The purpose of this paper is to begin the study of that part of the prime spectrum of $${\mathcal O}_q(M_n(k))$$ consisting of ideals containing $$D$$, about which the earlier work can give no information.
The main result of this paper is a quantum analogue of the first fundamental theorem of invariant theory. It states that, for $$t$$ an integer between $$0$$ and $$n-1$$, the ideal of $${\mathcal O}_q(M_n(k))$$ generated by the $$(t+1)\times(t+1)$$ quantum minors of $${\mathcal O}_q(M_n(k))$$ is completely prime. The ideas developed for its proof are likely to find further application in the analysis of $${\mathcal O}_q(M_n(k))$$, the main technical tool being the construction of a suitable vector space basis of $${\mathcal O}_q(M_n(k))$$, an analogue of the standard basis in the commutative case, such that $${\mathcal O}_q(M_n(k))$$ can be viewed as a “noncommutative algebra with straightening law” with respect to this basis.

##### MSC:
 16P40 Noetherian rings and modules (associative rings and algebras) 16D25 Ideals in associative algebras 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 16W35 Ring-theoretic aspects of quantum groups (MSC2000) 13C40 Linkage, complete intersections and determinantal ideals
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