Generators and relations for 3-graded Lie algebras
The main result of this paper can be viewed as being for Kantor-Koecher- Tits (KKT-) algebras of a Jordan pair spanned by a standard grid an analogue of the familiar construction of finite-dimensional semisimple Lie algebras using generators and relations based on root bases for root systems. (KKT-algebras are 3-graded Lie algebras \({\mathcal L}_ 1\oplus{\mathcal L}_ 0\oplus {\mathcal L}_{-1}\) for which \({\mathcal L}_ 0=[{\mathcal L}_ 1{\mathcal L}_{-1}]\) and \(\{x\in{\mathcal L}_ 0: [x{\mathcal L}_ 1]=[x{\mathcal L}_{-1}]=0\}=0\). (\(({\mathcal L}_ 1,{\mathcal L}_{-1})\) is a Jordan pair with respect to \(\{x^ \sigma y^{-\sigma} z^ \sigma\}=[[x^ \sigma y^{-\sigma}] z^ \sigma]\) for \(\sigma=\pm 1\) and \(x^ \sigma \in L_ \sigma\). A standard grid in a Jordan pair is a family of idempotents satisfying certain multiplication rules.) In this construction of KKT-algebras a root system \(R\) is allowed to lie in an arbitrary real inner product space, and root bases are replaced by grid bases \(B\), which are vector space bases contained in \(R\) such that every root in \(R\) is an alternating sum of roots in \(B\) satisfying another technical property. The special case when \(R\) is finite is stated as a Corollary. There is also an application to \(L^*\)-algebras.
scientific article; zbMATH DE number 168176