Cross-projective representations of pairs of anticommutative algebras,   alloys and finite-dimensional irreducible representations of some   infinite-dimensional Lie algebras
  The article is devoted to some ``strange'' phenomena of representation theory and their interrelations. Cross-projective representations of pairs of anticommutative algebras, alloys, their universal envelopping Lie algebras and their representations, quaternary algebras and their alloyability are discussed. Considered examples allow to conclude that new representations have some intriguing features (continuous moduli of finite-dimensional irreducible representations, sophisticated Clebsch-Gordan coefficient calculus, etc.). 
Denis V.Juriev