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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.

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
Full Text: DOI
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