The moment map for the variety of Leibniz algebras (Q6608785)

The paper under review explores the moment map \(m : PV_n \to i u(n)\) associated with the action of \(GL(n)\) on the space \(V_n = \otimes^2 (C^n)^* \otimes C^n\). The authors focus on the functional \(F_n = \|m\|^2\), restricting their attention to its behaviour on the projectivized varieties of \(n\)-dimensional Leibniz algebras \(L_n\) and symmetric Leibniz algebras \(S_n\). They derive some significant results regarding the extrema and critical points of \(F_n\) on these varieties.\N\NThe paper begins by recalling the moment map's definition in the context of Leibniz algebras, a non-anticommutative generalization of Lie algebras. It emphasizes the role of the invariant Hermitian structure and how the critical points of \(F_n\) can be characterized in terms of the derivation algebra. The authors prove that for any critical point \([\mu]\) in \(S_n\), the eigenvalues of the associated derivation matrix are nonnegative and rational.\N\NOne of the major contributions is the classification of critical points for \(n = 2, 3\). For \(n = 2\), the authors show that every two-dimensional symmetric Leibniz algebra corresponds to a critical point. For \(n = 3\), they demonstrate the existence of symmetric Leibniz algebras that do not arise as critical points of \(F_n\). Furthermore, they describe how the minimal and maximal values of \(F_n\) are achieved. The minimum is attained at semisimple Lie algebras, while the maximum occurs at a specific direct sum of the two-dimensional non-Lie symmetric Leibniz algebra and a trivial algebra.\N\NThe paper employs advanced techniques from algebraic geometry, symplectic geometry, and representation theory to extend Lauret's work on Lie algebras to Leibniz algebras. In doing so, it generalizes some known results about the moment map and provides a deeper understanding of the stratification of algebra varieties.\N\NOverall, the paper is a valuable contribution to the field of algebraic geometry and representation theory. It connects classical moment map theory with modern investigations into the structure of non-Lie algebras and their geometric properties, offering a framework for further exploration.