Leibniz Algebras with Invariant Bilinear Forms and Related Lie Algebras

Abstract

In ([11]), we have studied quadratic Leibniz algebras that are Leibniz algebras endowed with symmetric, nondegenerate, and associative (or invariant) bilinear forms. The nonanticommutativity of the Leibniz product gives rise to other types of invariance for a bilinear form defined on a Leibniz algebra: the left invariance, the right invariance. In this article, we study the structure of Leibniz algebras endowed with nondegenerate, symmetric, and left (resp. right) invariant bilinear forms. In particular, the existence of such a bilinear form on a Leibniz algebra 𝔏 gives rise to a new algebra structure ☆ on the underlying vector space 𝔏. In this article, we study this new algebra, and we give information on the structure of this type of algebras by using some extensions introduced in [11]. In particular, we improve the results obtained in [22].

Key Words:

• Double extension
• Left (resp. Right) invariant bilinear form
• Leibniz algebra
• Levi–Civita product
• Pseudo-metric on Lie algebra
• T*-extension

2010 Mathematics Subject Classification:

• 17A32
• 17B30
• 17B40
• 17A60