A Lie–Rinehart Algebra with No Antipode

Krähmer, Ulrich; Rovi, Ana

doi:10.1080/00927872.2014.896375

Cited by 21 publications

(16 citation statements)

References 15 publications

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“…An important milestone in the development of these algebraic structures is Rinehart's work [Rin63] in which he gives the structure of their universal enveloping algebra (see [Rin63, Section 2]), generalizing the construction of the universal enveloping algebra of a Lie algebra. In fact, as proved in [KR13], A will not carry a right V (A, L)-module structure in general. We will only be considering right (A, L)-module structures (and connections) on A.…”

Section: Lie-rinehart Algebrasmentioning

confidence: 97%

“…In [BS04], an example of a full Hopf algebroid is given that does not satisfy the axioms of [Lu96]; it is however unknown whether all Hopf algebroids in the sense of [Lu96] are full. Moreover, until [KR13] it had been an open question whether Hopf algebroids, in the sense of [BS04] or in the sense of [Lu96], were equivalent to left Hopf algebroids. It turns out that the universal enveloping algebra of a Lie-Rinehart algebra (a fundamental example of a left Hopf algebroid, see [KK10, Example 2]) will not carry an antipode in general.…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by [KR13], in the present note we give further examples of Hopf algebroids without an antipode, i.e., left Hopf algebroids which are neither full nor satisfy the axioms of [Lu96]. In these examples we consider Jacobi algebras, a generalization of Poisson algebras first introduced in a differential geometric context in [Kir76,Lic78], and construct quotients of a canonical Lie-Rinehart algebra associated to them (see Lemma 3.4 in Section 3.1).…”

Section: Introductionmentioning

confidence: 99%

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Hopf algebroids associated to Jacobi algebras

Rovi

2014

Int. J. Geom. Methods Mod. Phys.

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Section: Lie-rinehart Algebrasmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hopf algebroids associated to Jacobi algebras

Rovi

2014

Int. J. Geom. Methods Mod. Phys.

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“…Concretely, A has a canonical left (A, L)-module structure but it does not hold a canonical right (A, L)-module structure. See [7] for a characterization of right (A, L)-module structures and see [11] for a concrete example.…”

Section: Actions and Semidirect Product Of Lie-rinehart Algebrasmentioning

confidence: 99%

Universal central extensions of Lie–Rinehart algebras

2018

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In this paper we study the universal central extension of a Lie-Rinehart algebra and we give a description of it. Then we study the lifting of automorphisms and derivations to central extensions. We also give a definition of a non-abelian tensor product in Lie-Rinehart algebras based on the construction of Ellis of non-abelian tensor product of Lie algebras. We relate this non-abelian tensor product to the universal central extension.2010 Mathematics Subject Classification. 17B55.

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“…A Lie-Rinehart algebra (L, A) can be viewed as a Lie algebra L, which is simultaneously an A-module, where A is an associative and commutative algebra, in such a way that both structures are related in an appropriate way. For more details about the history and the developments of Lie-Rinehart algebras, see [13,14,15,19,20,21,29] and references cited therein. for all D 1 , D 2 ∈ Der φ (A).…”

Section: Introductionmentioning

confidence: 99%

On the structure of split regular Hom-Lie–Rinehart algebras

2020

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The aim of this paper is to study the structures of split regular Hom-Lie Rinehart algebras. Let (L, A) be a split regular Hom-Lie Rinehart algebra. We first show that L is of the form L = U + [γ]∈Γ/∼ I [γ] with U a vector space complement in H and I [γ] are well described ideals of L satisfying I [γ] , I [δ] = 0 if I [γ] = I [δ] . Also, we discuss the weight spaces and decompositions of A and present the relation between the decompositions of L and A. Finally, we consider the structures of tight split regular Hom-Lie Rinehart algebras.There is a growing interest in twisted algebraic structures or Hom-algebraic structures defined for classical algebras and Lie algebroids as well. Hom-algebras were first introduced in the Lie algebra setting [16] with motivation from physics though its origin can be traced back in earlier literature such as [18]. In [23], Makhlouf and Silvestrov introduced the definition of Hom-associative algebras, where the associativity of a Hom-algebra is twisted by an endomorphism. Recently, Mandal and Mishra introduced the notion of Hom-Lie Rinehart algebras in [24], which is a generalization of Lie-Rinehart algebras.The class of the split algebras is specially related to addition quantum numbers, graded contractions and deformations. For instance, for a physical system which displays a symmetry of L, it is interesting to know in detail the structure of the split decomposition because its roots can be seen as certain eigenvalues which are the additive quantum numbers characterizing the state of such system. Determining the structure of split algebras will become more and more meaningful in the area of research in mathematical physics. Recently,in [1,2,4,5,6,7,8,9,10,11,12], the structure of different classes of split algebras have been determined by the techniques of connections of roots.In the present paper we introduce the class of split regular Hom-Lie Rinehart algebras as the natural extension of the one of split Lie-Rinehart algebras and so of split regular Hom-Lie algebras, and study its tight structures based on some work in [1] and [3]. In section 2, we establish the preliminaries on split regular Hom-Lie Rinehart algebras theory. In sections 3 and 4, we develop techniques of connections of roots and weights for split Hom-Lie Rinehart algebras respectively. In section 5, we study the structures of tight split regular Hom-Lie Rinehart algebras. PreliminariesIn this section, we start by recalling the definition of Hom-Lie Rinehart algebras, then we introduce the notions of roots and weights of split Hom-Lie Rinehart algebras. Throughout the paper, all algebraic systems are supposed to be over a field k. Definition 2.1. ([25]) Let A be an associative commutative algebra and φ : A → A an algebra endomorphism. A φ-derivation on A is a linear map D : A → A satisfying D(ab) = φ(a)D(b) + D(a)φ(b), (2. 1) for all a, b ∈ A. The set of all φ-derivations on A is denoted by Der φ (A).Remark 2.2. The triple (Der φ (A), [·, ·] φ , ψ φ ) is a Hom-Lie algebra, where the bracket [·, ·] φ and the ...

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A Lie–Rinehart Algebra with No Antipode

Abstract: The aim of this note is to communicate a simple example of a Lie-Rinehart algebra whose enveloping algebra is not a Hopf algebroid, neither in the sense of Böhm and Szlachányi, nor in the sense of Lu.

Cited by 21 publications

References 15 publications

Hopf algebroids associated to Jacobi algebras

Hopf algebroids associated to Jacobi algebras

Universal central extensions of Lie–Rinehart algebras

On the structure of split regular Hom-Lie–Rinehart algebras

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