Abstract
We define quantum determinants in quantum matrix algebras related to pairs of compatible braidings. We establish relations between these determinants and the so-called column and row determinants, which are often used in the theory of integrable systems. We also generalize the quantum integrable spin systems using generalized Yangians related to pairs of compatible braidings. We demonstrate that such quantum integrable spin systems are not uniquely determined by the “quantum coordinate ring” of the basic space \(V\). For example, the “quantum plane” \(xy=qyx\) yields two different integrable systems: rational and trigonometric.
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Notes
The term even means that the \(R\)-skew-symmetric algebra \( \mathsf{\Lambda} _R(V)\) has a finite number of nontrivial homogenous components and the highest nontrivial component \( \mathsf{\Lambda} _R^m(V)\) is one-dimensional. In this case, we say that \(R\) is of rank \(m\).
We note that the operators \(PR\), where \(P\) is a flip, are subject to the so-called quantum Yang–Baxter equation and are usually called \(R\)-matrices.
In this section, we mainly consider Hecke symmetries. The corresponding results and formulas for involutive symmetries can be obtained by setting \(q=1\).
If the rank of a symmetry \(R\) is two, then we can recover the symmetry \(R\) by knowing \( \mathrm{u} \) and \( \mathrm{v} \). All pairs \(( \mathrm{u} , \mathrm{v} )\) yielding such symmetries were classified in [2].
We emphasize that this property is absent if \(F\ne R\).
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Acknowledgments
The authors are indebted to Vladimir Rubtsov for the elucidating discussions.
Funding
The research of P. A. Saponov was supported in part by the Russian Foundation for Basic Research (Grant No. 19-01-00726_a).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2021, Vol. 207, pp. 261-276 https://doi.org/10.4213/tmf10043.
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Gurevich, D.I., Saponov, P.A. Determinants in quantum matrix algebras and integrable systems. Theor Math Phys 207, 626–639 (2021). https://doi.org/10.1134/S004057792105007X
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DOI: https://doi.org/10.1134/S004057792105007X