Remarks on generalized Lie algebroids and related concepts

Abstract

We prove that "generalized Lie algebroid", a geometric object which appeared recently in the literature, is a misconception.Comment: 9 page

Generalized Lie Algebroids - Examples by Distinguished Lie Algebroids with Applications to Optimal Control

“…Also, the Jacobi identity holds for this bracket. We remark that if ρ This section is devoted to detail that Theorem 3.1 in [15] is based on a completely false assumption and so is not valid. Indeed, component-wise composition of vector bundle morphism has no sense as a new vector bundle morphism; and this is their false assumption.…”

“…Finally, by notations used in [15] and gathering above discussions, we derive the following statement.…”

“…Therefore, the proof of Theorem 3.1 from [15] is based on a misconception and is not legal. It breaks Theorem 1.3 down as the only statement in [15].…”

“…So, the affirmation "...... the author's "generalized Lie algebra" is actually a particular case of a Lie pseudoalgebra (Lie-Rinehart algebra) mentioned in the introduction." from [15], pag. 5 is completely false.…”

“…Section 4, is devoted to show that the proof of Theorem 3.1 (as the only result) from [15] is a misconception and so why the authors of [15] have wrong ratiocination.…”