Hopf algebroids associated to Jacobi algebras

Abstract

We give examples of Lie-Rinehart algebras whose enveloping algebra is not a full Hopf algebroid in the sense of Bohm and Szlachanyi. We construct these examples as quotients of a canonical Lie-Rinehart algebra over a Jacobi algebra which does admit an antipode

Lie Algebroids in the Loday-Pirashvili Category

“…Since a very rich class of examples of Hopf algebroids [3,13,22] is the enveloping algebra of Lie-Rinehart algebras, following the argument given by Loday and Pirashvili in [19], we would expect a similar relation between the enveloping algebra of a Leibniz algebroid and Hopf algebroids in LM. However this generalisation of [19] requires not only the definition of a correct notion of enveloping algebra of a Leibniz algebroid (not yet defined in the literature), but also the construction of a functor from the category of Leibniz algebroids to the category of Lie-Rinehart algebras objects in LM, which goes beyond the goals of this paper.…”

“…• Similarly, let A be a Jacobi algebra with bracket {−, −} J , and let J 1 (A) be its 1-jet space. Then the pair (A, J 1 (A)) is a Lie-Rinehart algebra (see [22] for more details), and the map j : A → J 1 (A) given by a → j 1 (a) is a Lie algebra object in LM with right J 1 (A)-action on A given by a ⊗ b • j 1 (c) → b • {a, c} J for all a, b, c ∈ A.…”

Hopf algebroids and twists for quantum projective spaces

“…Indeed there are situations where an antipode does not exist at all. We refer to [15] for one such an example.…”

“…In contrast, for a bialgebroid H, the space H copop is not a bialgebroid in the same sense as H. A first definition of an antipode in this setting appeared in [13] and later in [1,2], while an abstract category-theoretical approach that does not refer to an antipode can be found in [7,16]. The latter definition turns out to be weaker then the former since a Hopf algebroid with an (invertible) antipode in the sense of [1,2] is also a Hopf algebroid in the sense of [16] (see Proposition 2.7 below), but the other way around is not true in general; there are bialgebroids that satisfy the requirement of [16] but they do not admit any antipode [15]. In [1] the notion of twist of an antipode is introduced.…”

Jacobi and Poisson algebras

“…It is therefore natural to expect that Jacobi algebras will play an important role in all fields enumerated above. For further details on the study of Jacobi algebras from geometric view point we refer to [20,22,23,38,45].…”