A Lie-Rinehart algebra with no antipode

Kraehmer, Ulrich; Rovi, Ana

doi:10.48550/arxiv.1308.6770

Cited by 3 publications

(5 citation statements)

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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: Definition 29 ([8]) a Right Lie-rinehartmentioning

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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Section: Definition 29 ([8]) a Right Lie-rinehartmentioning

confidence: 99%

Universal central extensions of Lie–Rinehart algebras

2018

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“…Finally, by a Hopf algebroid we mean left rather than full Hopf algebroid, so there is in general no antipode [KR13]: Definition 5.4 ([Sch00]). A Hopf algebroid is a bialgebroid with bijective Galois map…”

Section: Bialgebroids and Hopf Algebroidsmentioning

confidence: 99%

Cyclic homology arising from adjunctions

Kowalzig¹,

Kraehmer²,

Slevin³

2015

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Given a monad and a comonad, one obtains a distributive law between them from lifts of one through an adjunction for the other. In particular, this yields for any bialgebroid the Yetter-Drinfel'd distributive law between the comonad given by a module coalgebra and the monad given by a comodule algebra. It is this self-dual setting that reproduces the cyclic homology of associative and of Hopf algebras in the monadic framework of Böhm and S ¸tefan. In fact, their approach generates two duplicial objects and morphisms between them which are mutual inverses if and only if the duplicial objects are cyclic. A 2-categorical perspective on the process of twisting coefficients is provided and the rôle of the two notions of bimonad studied in the literature is clarified.

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“…In [KP11, Proposition 3.11], it is proved that there exists an antipode on the universal enveloping algebra of (A, L) turning the left Hopf algebroid structure on V (A, L) into a full Hopf algebroid if and only if there exists a right V (A, L)-module structure on A. From [KR13], it follows that the left Hopf algebroid V (A, L) is not, in general, a full one. However, examples where V (A, L) admits an antipode do exist: e.g.…”

Section: Right (A L)-module Structures and Connections On Amentioning

confidence: 99%

“…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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