Lie theory for symmetric Leibniz algebras
Abstract
Abstract Lie algebras and groups equipped with a multiplication $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>μ</mml:mi></mml:math> satisfying some compatibility properties are studied. These structures are called symmetric Lie $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>μ</mml:mi></mml:math> -algebras and symmetric $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>μ</mml:mi></mml:math> -groups respectively. An equivalence of categories between symmetric Lie $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>μ</mml:mi></mml:math> -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 $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>μ</mml:mi></mml:math> -groups and finite dimensional symmetric Leibniz algebras.