\(W_m\)-algebras and fractional powers of difference operators (Q6583637)

Adler-Gel'fand-Dickey (AGD) flows were originally defined on the space of operators \(u_0(x) + \cdots + u_{m-2}(x) \partial^{m-2} - \partial^m\) where \(\partial = \frac{\partial}{\partial x}\) and the coefficients of \(u_i(x)\) are periodic functions. This space of operators has a quadratic Poisson structure (sometimes called the second Adler-Gel'fand-Dickey bracket). Its Poisson algebra is the classical \(W_m\)-algebra.\N\NThe author focuses on discretizations of ADG flows, a continuation on the work begun by her with \textit{A. Calini} in [Int. Math. Res. Not. 2022, No. 6, 4318--4375 (2022; Zbl 1490.53015)] and with \textit{A. Izosimov} in [Int. Math. Res. Not. 2023, No. 19, 17021--17059 (2023; Zbl 1547.17056)]. \N\NShe now defines a Poisson pencil associated to a lattice \(W_m\) algebra defined in the second cited reference. She proves that this is equal to the one previously defined by the author and \textit{J. Wang} in [Nonlinearity 26, 2515-- 2551 (2013; Zbl 1309.37056)]. The author then shows that a family of Hamiltonians defined by fractional powers of the difference operators commute with respect to both structures. She defines the kernel of one structure and constructs an integrable hierarchy in the sense of Liouville.