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Galois cohomology and forms of algebras over Laurent polynomial rings. (English) Zbl 1131.11070
The main idea of the paper under review is as follows. Suppose we are given a certain class of infinite dimensional Lie algebras \(L\) defined over an algebraically closed field \(k\) such that its centroid \(C=Ctd_k(L)\) (the subring of the ring of endomorphisms of \(L\) which consists of all elements commuting with the Lie bracket) can be identified with the ring \(R_n\) of Laurent polynomials in finitely many variables \(t_1^{\pm },\dots, t_n^{\pm }\). Then in many cases \(L\) can be viewed as a twisted form of a finite dimensional Lie algebra \({\mathfrak g}\) defined over \(k\) (i.e. for some finite Galois extension \(S/C\) there is an isomorphism of \(S\)-algebras \(L\otimes _CS\) and \({\mathfrak g}\otimes _kS\)). In such a situation, one can attach to \(L\) a torsor \(X_L\) whose isomorphism class lies in \(H^1_{\text{et}}(R_n, \operatorname{Aut}({\mathfrak g}))\). This is the case, for example, for an affine Kac-Moody Lie algebra (more precisely, for its derived algebra factored by the centre), where we have \(n=1\), or for a so-called extended affine Lie algebra (EALA) (more precisely, for its centreless core), see, e.g., B. N. Allison et al. [Mem. Am. Math. Soc. 603 (1997; Zbl 0879.17012)].
Taking this point of view, it is a natural question to ask whether Serre’s conjectures hold true in the above setting. If \(n=1\), Serre’s Conjecture I was established by the second author [C. R., Math., Acad. Sci. Paris 340, 633–638 (2005; Zbl 1078.14064)]. In the case \(n=2\), the authors discuss an analogue of Serre’s conjecture II and present several positive results as well as a counter-example. To get such a counter-example, some general construction is used. For an arbitrary \(n\), the authors associate to any \(n\)-tuple \(x\) of commuting elements \(x_1,\dots ,x_n\) of finite order in a reductive \(k\)-group \(G\) a certain cocycle \(\alpha (x)\in H^1(R_n,G)\) called a loop torsor. They study a certain invariant (Witt-Tits index) associated to such an \(x\); for \(n=2\) they also study another invariant with values in the Brauer group of \(R_n\). It turns out that the Brauer invariant classifies inner loop torsors but is not fine enough to distinguish arbitrary ones. This leads to the needed counter-example (an anisotropic form of type \(A_n\)).
On the positive side, they state a conjecture that any semisimple \(R_2\)-group without factors of type \(A\) the connecting map \(H^1(R_2,G)\to H^2(R_2,\mu )\) is bijective (\(\mu\) stands for the kernel of the simply connected covering of \(G\)). In particular, this would imply that for such a \(G\), all \(R_2\)-torsors are loop torsors. This conjecture is true for the groups of types \(G_2\), \(F_4\), \(E_8\), as well as for the special orthogonal and spinor groups of quadratic forms of rank at least 5 (the last case is a consequence of the classification results due to R. Parimala [Trans. Am. Math. Soc. 277, 569–578 (1983; Zbl 0514.13005)].
The authors then apply this approach to classification problems for some infinite dimensional algebras. After some generalities on twisted forms of algebras over rings (which are interesting by their own) they show that the approach described above applies to the study of the group of automorphisms of many important families of infinite dimensional Lie algebras (in particular, EALA’s). They discuss two examples: quantum tori and affine Kac-Moody Lie algebras. The last case is an instance of the so-called multi-loop algebras for which some general results are presented. In the case \(n=2\), the authors state a conjecture saying that outside of type \(A\), double loop algebras are completely determined by their Witt-Tits index.

MSC:
11R34 Galois cohomology
14L15 Group schemes
17B56 Cohomology of Lie (super)algebras
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
20G10 Cohomology theory for linear algebraic groups
20G15 Linear algebraic groups over arbitrary fields
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