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

TL;DR

This paper constructs a family of principal quasi-determinants for RE-algebras associated with the Drinfeld R-matrix, generalizing Toda system integrals and clarifying RE-algebras' role as quantum homogeneous spaces.

Contribution

It introduces a new family of quasi-determinants for RE-algebras that generalize Toda system integrals and elucidates their significance as quantum homogeneous spaces.

Findings

Found principal quasi-determinants for RE-algebras with Drinfeld R-matrix.

Showed these quasi-determinants commute and their product equals the quantum determinant.

Linked RE-algebras to quantum homogeneous spaces and potential quantum field theories.

Abstract

In 1991, Gelfand and Retakh embodied the idea of a noncommutative Dieudonne determinant in the case 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.