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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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