Durch graduierte Lie-Algebren definierte Paar-Algebren (Pair algebras defined by graded Lie algebras)
A pair \({\mathcal A}=(A^-,A^+)\) of vector spaces over a field K with trilinear mappings (x,u,y)\(\to \{xuy\}_{\epsilon}\) from \(A^{\epsilon}\times A^{-\epsilon}\times A^{\epsilon}\) into \(A^{\epsilon}\) is called a pair algebra if \(\{xu\{yvz\}_{\epsilon}\}_{\epsilon}- \{yv\{xuz\}_{\epsilon}\}_{\epsilon}\}=\{\{xuy\}_{\epsilon}vz\}_{\epsilon}- \{y\{uxv\}_{-\epsilon}z\}_{\epsilon}\) for all \(x,y,z\in A^{\epsilon}\), \(u,v\in A^{-\epsilon},\epsilon =\pm\). With a pair algebra \({\mathcal A}\) there is associated a certain graded Lie algebra \({\mathcal S}({\mathcal A})=\oplus S({\mathcal A})_ r\) (r\(\in {\mathbb{Z}})\), known as the standard algebra of A, in which S(\({\mathcal A})_{-1}=A^-\) and S(\({\mathcal A})_ 1=A^+\). The author's investigation includes a characterization of those pair algebras A for which S(\({\mathcal A})\) is a Kac-Moody algebra as being pair algebras having a nondegenerate (left- and right-) invariant bilinear form and for which the inner derivation algebra is completely reducible. More detailed results are obtained when S(\({\mathcal A})\) is an affine Lie algebra.
scientific article; zbMATH DE number 3908642