The symplectic groupoid for Adler-Gelfand-Dikii Poisson structure

TL;DR

This paper constructs the symplectic groupoid for the Adler-Gelfand-Dikii Poisson structure linked to classical Lie groups, establishing its Morita equivalence with a quasi-symplectic groupoid related to monodromy-preserving curve spaces.

Contribution

It introduces a construction of the symplectic groupoid for the Adler-Gelfand-Dikii Poisson structure and proves its Morita equivalence to a quasi-symplectic groupoid associated with monodromy.

Findings

Construction of the symplectic groupoid for the Adler-Gelfand-Dikii structure

Proof of Morita equivalence with a quasi-symplectic groupoid

Connection between Poisson structures and monodromy-preserving curve spaces

Abstract

The Adler-Gelfand-Dikii Poisson structure arises naturally in the study of $n$-th order differential operators on the circle and plays a central role in Poisson geometry and integrable systems. Let $G$ be one of the Lie groups $\mathrm{PSL}(n)$, $\mathrm{PSp}(n)$ (for even $n$), or $\mathrm{PSO}(n)$ (for odd $n$). In this paper, we construct the symplectic groupoid integrating the Adler-Gelfand-Dikii Poisson structure associated to $G$ and prove that it is Morita equivalent to the quasi-symplectic groupoid integrating the Dirac structure on $Y_n(\mathbf{C})$, where $Y_n(\mathbf{C})$ denotes the quotient of the space of quasi-periodic non-degenerate curves by homotopies preserving the monodromy.