Imbedding of pair algebras into Lie algebras

AbstractFrom a pair algebra, i.e. a pair A=(A−,A+) of vector spaces equipped with trilinear mappings Aε×A−ε×Aε→Aε (ε=±) satisfying a certain identity we construct a universal enveloping Z-graded Lie algebra P(A), exhibit some of its structural properties (ideals, homomorphisms, derivations) and give, under certain restrictions on A, a necessary and sufficient condition for P(A) to be finite dimensional in terms of A