Partial Differential Equations in Applied Mathematics (Sep 2024)

RE-algebras, quasi-determinants and the full Toda system

  • Dmitry V. Talalaev

Journal volume & issue
Vol. 11
p. 100898

Abstract

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In 1991, Gelfand and Retakh embodied the idea of a noncommutative Dieudonne determinant for a generating matrix of RTT algebra, namely, they found a representation of the quantum determinant of RTT algebra in the form of a product of principal quasi-determinants. In this note we construct an analogue of the above statement for the RE-algebra corresponding to the Drinfeld R-matrix for the order n=2,3. Namely, we have found a family of quasi-determinants that are principal with respect to the antidiagonal, commuting among themselves, whose product turns out to be the quantum determinant of this algebra. This family generalizes the construction of integrals of the full Toda system due to Deift et al. for the quantum case of RE-algebras. In our opinion, this result also clarifies the role of RE-algebras as a quantum homogeneous spaces and can be used to construct effective quantum field theories with a boundary.

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