Cohomology of restricted Lie–Rinehart algebras and the Brauer group

Dokas, Ioannis

doi:10.1016/j.aim.2012.08.003

Cited by 18 publications

(17 citation statements)

References 15 publications

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“…Definition 8.1. [Do,Definition 1.7] A restricted Lie-Rinehart algebra (A, L, (−) [p] ) over a commutative k-algebra A, is a Lie-Rinehart algebra over A such that (a) (L, (−) [p] ) is a restricted Lie algebra over k, (b) the anchor map is a restricted Lie homomorphism, and (c) the following relation holds:…”

Section: Connection With Restricted Lie-rinehart Algebrasmentioning

confidence: 99%

Restricted Poisson algebras

Bao

Zhang

2017

Pacific J. Math.

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Section: Connection With Restricted Lie-rinehart Algebrasmentioning

confidence: 99%

Restricted Poisson algebras

Bao

Zhang

2017

Pacific J. Math.

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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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“…In Section 2, we recall some basic definitions of restricted Lie-Rinehart algebras. In Section 3, we introduce the definition of restrictable Lie-Rinehart algebras, which is by far more tractable than that of a restricted Lie-Rinehart algebras in [6]. Moreover, we obtain some properties of p-mappings and restrictable Lie-Rinehart algebras.…”

Section: Introductionmentioning

confidence: 99%

“…In recent years the study of restricted Lie-Rinehart obtained some important results. In [6], I. Dokas introduced the notion of restricted Lie-Rinehart algebras and constructed its restricted enveloping algebra. As a natural generalization of a restricted Lie algebra, it seems desirable to investigate the possibility of establishing a parallel theory for restricted Lie-Rinehart algebras.…”

Section: Introductionmentioning

confidence: 99%

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Restricted and quasi-toral restricted Lie-Rinehart algebras

Sun

Chen

2015

Open Mathematics

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Abstract:In this paper, we introduce the definition of restrictable Lie-Rinehart algebras, the concept of restrictability is by far more tractable than that of a restricted Lie-Rinehart algebra. Moreover, we obtain some properties of p-mappings and restrictable Lie-Rinehart algebras. Finally, we give some sufficient conditions for the commutativity of quasi-toral restricted Lie-Rinehart algebras and study how a quasi-toral restricted Lie-Rinehart algebra with zero center and of minimal dimension should be.

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Cohomology of restricted Lie–Rinehart algebras and the Brauer group

Cited by 18 publications

References 15 publications

Restricted Poisson algebras

Restricted Poisson algebras

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

Restricted and quasi-toral restricted Lie-Rinehart algebras

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