Algebraic Inverses on Lie Algebra Comultiplications

In this note, we investigate algebraic loop structures and inverses of elements of a set of all homomorphisms of Lie algebras with a binary operation derived from a Lie algebra comultiplication. As a symmetry phenomenon, we show that if l ( 1 ) c and r ( 1 ) c are the left and right inverses of the identity 1 : L → L on a free graded Lie algebra L , respectively, based on the Lie algebra comultiplication ψ c : L → L ⊔ L , then we have l ( 1 ) = l ( 1 ) c and r ( 1 ) = r ( 1 ) c , where c : L → L ⊔ L is a commutator.