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Minor identities for quasi-determinants and quantum determinants.(English)Zbl 0829.15024

Matrices \(T\) with noncommutative entries, obeying a relation of the form \(RT_1 T_2 = T_2T_1R\), where the matrix \(R\) is a solution of the Yang-Baxter equation, are considered. The entries of the monodromy matrix \(T\) may be regarded as the generators of an associated algebra subject to the above relation.
The aim of the article is to demonstrate that the quasi-determinants of Gelfand and Retakh can be used to describe the algebraic relations satisfied by the quantum minors of a monodromy matrix. Therefore identities satisfied by quasi-minors of the generic noncommutative matrix are studied.
Then quantum determinantal identities are derived by specializing them to a matrix \(T\). Only the matrix \(T\) of the generators of the quantum group \(A_q (GL)_n\) is considered (some techniques are also applicable to the case of \(Y(g \ell_n))\).
Reviewer: V.Burjan (Praha)

MSC:

15A90 Applications of matrix theory to physics (MSC2000)
15A15 Determinants, permanents, traces, other special matrix functions
81U40 Inverse scattering problems in quantum theory

Cite

References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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