Fourier–Mukai transform on Weierstrass cubics and commuting differential operators

Abstract

In this article, we describe the spectral sheaves of algebras of commuting differential operators of genus one and rank two with singular spectral curve, solving a problem posed by Previato and Wilson. We also classify all indecomposable semi-stable sheaves of slope one and ranks two or three on a cuspidal Weierstraß cubic.

Torsion Free Sheaves on Weierstrass Cubic Curves and the Classical Yang–Baxter Equation

“…for a local function f as above. Arguing as in the proof of Proposition 3.6, we get a short exact sequence (19). Then we get:…”

“…Here, the Lie algebra A p is the germ of the sheaf A at the point p and A p is its completion. An interested reader might look for a more detailed exposition in [19,Section 1.7], where the Beauville-Laszlo construction occurred in another setting. The following sequence of vector spaces (which is a version of the Mayer-Vietoris sequence)…”

“…Let us choose local coordinates in a neighbourhood of the point (p, p) ∈Ȇ ×Ȇ, following (59). According to (19), we can write: ρ(x 1 ,…”

Centralizers of Rank One in the First Weyl Algebra

“…Interested reader will also benefit by looking at lectures by Emma Previato [35] and at the paper [43] by George Wilson as well as papers [38,39,42], and [2].…”

Cohen–Macaulay modules over the algebra of planar quasi–invariants and Calogero–Moser systems

“…In the one‐dimensional case, the isomorphism () is due to Mumford [33, Section 2]. For partial differential operators, we follow the exposition in [9, Theorem 1.14]. The key point is the following isomorphism of left ‐modules: where we take the right action on and the usual right action on of the algebra .…”