Manin triples and bialgebras of Left-Alia algebras associated to invariant theory

A left-Alia algebra is a vector space together with a bilinear map satisfying symmetric Jocobi identity. Motivated by invariant theory, we first construct a class of left-Alia algebras induced by twisted derivations. Then, we introduce the notion of Manin triples and bialgebras of left-Alia algebras. Via specific matched pairs of left-Alia algebras, we figure out the equivalence between Manin triples and bialgebras of left-Alia algebras.