Lie theory for symmetric Leibniz algebras

Abstract

Abstract Lie algebras and groups equipped with a multiplication $$\mu $$μ satisfying some compatibility properties are studied. These structures are called symmetric Lie $$\mu $$μ-algebras and symmetric $$\mu $$μ-groups respectively. An equivalence of categories between symmetric Lie $$\mu $$μ-algebras and symmetric Leibniz algebras is established when 2 is invertible in the base ring. The second main result of the paper is an equivalence of categories between simply connected symmetric Lie $$\mu $$μ-groups and finite dimensional symmetric Leibniz algebras.