We define hom-Lie algebroids, a definition that may seem cumbersome at first, but which is justified, first, by a one-to-one corespondence with hom-Gerstenhaber algebras, a notion that we also introduce, and several examples, including hom-Poisson structures

Abstract. We construct the universal enveloping algebra of a Hom-Lie algebra and endow it with a Hom-Hopf algebra structure. We discuss group-like elements that we see as a Hom-group integrating the initial Hom-Lie algebra.

The introduction of automated deduction systems in secondary schools faces several bottlenecks. Beyond the problems related with the curricula and the teachers, the dissonance between the outcomes of the geometry automated theorem provers and the normal practice of conjecturing and proving in schools is a major barrier to a wider use of such tools in an educational environment.Since the early implementations of geometry automated theorem provers, applications of artificial intelligence methods, synthetic provers based on inference rules and using forward chaining reasoning are considered to be best suited for education proposes.Choosing an appropriate set of rules and an automated method that can use those rules is a major challenge. We discuss one such rule set and its implementation using the geometry deductive databases method (GDDM). The approach is tested using some chosen geometric conjectures that could be the goal of a 7th year class (≈12-year-old students). A lesson plan is presented, its goal is the introduction of formal demonstration of proving geometric theorems, trying to motivate students to that goal.

We obtain a bijection between the set of equivalent classes of invariant star products on a non-degenerate triangular finite dimensional Lie bialgebra over the formal power series ring and the set , working in the framework developed by Etingof–Kazhdan for the quantization of Lie bialgebras. Two of the corresponding triangular Hopf algebras over the ring are isomorphic if and only if the invariant star products defining them are equivalent. Therefore, when t = ℏ, we obtain a set of triangular Hopf quantized universal enveloping algebras which can also be seen as quantizations of the deformation algebra . Additionally, two of them are isomorphic if and only if the above invariant star products are equivalent.