Lie algebroids arising from simple group schemes

A classical construction of Atiyah assigns to any (real or complex) Lie group G, manifold M and principal homogeneous G-space P over M, a Lie algebroid over M ([1]). The spirit behind our work is to put this work within an algebraic context, replace M by a scheme X and G by a “simple” reductive group scheme G over X (in the sense of Demazure-Grothendieck) that arise naturally with an attached torsor (which plays the role of P) in the study of Extended Affine Lie Algebras (see [9] for an overview). Lie algebroids in an algebraic sense were also considered by Beilinson and Bernstein. We will explain how the present work relates to theirs.