The classification of locally finite split simple Lie algebras (Q2710293)

A complete classification of the infinite dimensional locally finite split simple Lie algebras is given, over a field \(k\) of characteristic zero. Whereas in the case of countable dimension the traditional approach of choosing a root base and associating a Dynkin diagram to a root decomposition works fine, in the case of uncountable dimension there exists no root base and hence no Dynkin diagram. The authors overcome this problem by using, among other techniques, a generalized root basis. They follow, in principle, the classical approach, like in the finite-dimensional case. To some extent there also exist different approaches using Jordan theory.NEWLINENEWLINENEWLINEHere is a short summary of the very well written paper: in the first section the important subalgebras of the infinite matrix Lie algebra Lie \({\mathfrak{gl}}(J,k)\) are presented: Lie \({\mathfrak{sl}}(J,k)\), Lie \({\mathfrak{o}}(J,J,k)\), Lie \({\mathfrak{sp}}(J,k)\). They are simple algebras and they will exhaust all infinite dimensional locally finite split simple Lie algebras. In the second section, the results needed on locally finite root systems of simple type are summarized. Then the classification of these root systems follows. For each infinite cardinality \(J\) there exist four types of these root systems, \(A_J,B_J,C_J\) and \(D_J\), and each root system of the above type is isomorphic to one of these four root systems. Section four presents the extension theorem needed for the transfer of the classification from root systems to Lie algebras. In contrast to the finite-dimensional case, the two root systems \(B_J\) and \(D_J\) now both correspond to one Lie algebra: Lie \({\mathfrak o}(J,J,k)\). The next two sections finish the proof by showing that the three Lie algebras from above are really non-isomorphic. Sections seven and eight discuss the so called \(3\)-cores of locally finite root systems and the determination of all Hermitian forms of the above Lie algebras.