Infinite dimensional Lie algebras and descent theory (Q2836965)

In this short monograph, various interesting results concerning the extended affine Lie algebras and their relation to torsors under algebraic groups over the Laurent polynomial ring \(R\) in two variables over an algebraically closed field are obtained. The main objective of the work is the classification of \(R\)-torsors under the classical groups and sufficiently high rank. More specifically, the classification is obtained for the inner and outer types \(A\), as well as the outer types \(B,C,D\). This enables the author to deduce structural properties on the extended affine Lie algebras, as well as to provide an answer in the affirmative to a variant of the Serre Conjecture II concerning the ring \(R\), stating that a smooth \(R\)-torsor under a semisimple connected \(R\)-group of types \(B,C,D\) and sufficiently high rank is necessarily trivial. The main theorem of the memoir, concerning the étale cohomology, states that given a \(R_{2}\)-scheme \(G\) in a semisimple simply connected group of classical type, then \(H^{1}_{\text{ét}}(R_{2},G)=1\) if the rank is sufficiently high.NEWLINENEWLINEThe memoir is divided into five chapters and an Appendix. The first section is devoted to present the principal result, with a short discussion on infinite-dimensional Lie algebras. The structure of the work is outlined, and the necessary notations are presented.NEWLINENEWLINEIn Section 2 the required prerequisites are developed. These include some fundamental results on algebraic geometry and modules, Azumaya algebras and the Brauer group, the De Witt group and various reduction theorems, as well as some important cohomological properties. Involutions in Azumaya algebras are also revisited. In Section 3, the two conjectures first announced in [\textit{P. Gille} and \textit{A. Pianzola}, J. Pure Appl. Algebra 212, No. 4, 780--800 (2008; Zbl 1132.14042)] are revisited. The author introduces a loop algebra and considers two invariants (the Brauer invariant and the Witt-Tits index) that can be related to these algebras.NEWLINENEWLINEChapters four and five are devoted to the proof of the main theorem for the case of \(^{1}A_{n-1}\) and the orthogonal groups, as well as the types \(C_{n}\) and \(D_{n}\) and \(^{1}A_{n}\). The Gille spectral sequences for Azumaya algebras and schemes play a central role in the reasoning. In Chapter 6, the author shows that the previous results obtained for the classical groups allow a positive answer for the B conjecture of Gille and Pianzola. The final chapter shows the compatibility of the Morita theory in the context of the de Witt algebras and the spectral sequence of Gille.NEWLINENEWLINESummarizing, this work provides concise answers to various important structural results. Albeit being a quite technical work, the author succeeds in providing the necessary material for a self-contained exposition.