Hochschild products and global non-abelian cohomology for algebras. Applications

Agore, A. L.; Militaru, Gigel

doi:10.1016/j.jpaa.2016.06.013

Cited by 5 publications

(9 citation statements)

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“…e · e := a y, f · f := b y, e · f := z for some a, b ∈ k. We denote by (h(3, k), k) (a, b) this JJ algebra. If k = k 2 , we can easily see that, up to an isomorphism there are only three such algebras, namely (h(3, k), k) (0, 0) , (h(3, k), k) (1,0) and (h(3, k), k) (1,1) which appear in [5,Proposition 3.4] in different notational conventions though.…”

Section: Applications and Examplesmentioning

confidence: 99%

“…In Section 2 we deal with the main question addressed in this paper, namely the global extension (GE) problem, and the crossed product for JJ algebras is introduced as the object responsible for it. Introduced for Leibniz algebras in [20] and studied for associative/Poisson algebras in [1,2], the GE problem is a generalization of the classical Hölder's extension problem, and for JJ algebras consists of the following question:…”

Section: Introductionmentioning

confidence: 99%

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On a type of commutative algebras

Agore

Militaru

2015

Linear Algebra and its Applications

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We introduce some basic concepts for Jacobi-Jordan algebras such as: representations, crossed products or Frobenius/metabelian/co-flag objects. A new family of solutions for the quantum Yang-Baxter equation is constructed arising from any 3-step nilpotent Jacobi-Jordan algebra. Crossed products are used to construct the classifying object for the extension problem in its global form. For a given Jacobi-Jordan algebra A and a given vector space V of dimension c, a global non-abelian cohomological object GH 2 (A, V ) is constructed: it classifies, from the view point of the extension problem, all Jacobi-Jordan algebras that have a surjective algebra map on A with kernel of dimension c. The object GH 2 (A, k) responsible for the classification of co-flag algebras is computed, all 1 + dim(A) dimensional Jacobi-Jordan algebras that have an algebra surjective map on A are classified and the automorphism groups of these algebras is determined. Several examples involving special sets of matrices and symmetric bilinear forms as well as equivalence relations between them (generalizing the isometry relation) are provided.2010 Mathematics Subject Classification. 17C10, 17C55.

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Section: Applications and Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On a type of commutative algebras

Agore

Militaru

2015

Linear Algebra and its Applications

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“…In fact, [3,Lemma1.5], applied for the abelian case, shows that two extensions of P 0 by V 0 of the form (P ( , , θ) V, i V , π P ) and (P ( , , θ) V, i V , π P ) are equivalent if and only if = , = and θ ≡ θ . All these considerations lead to the following classification result that gives the decomposition of Ext(P 0 , V 0 ) as the co-product of all discrete cohomological groups.…”

Section: Remark 35mentioning

confidence: 99%

Metabelian Associative Algebras

Militaru¹

2015

Bull. Malays. Math. Sci. Soc.

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Metabelian algebras are introduced and it is shown that an algebra A is metabelian if and only if A is a nilpotent algebra having the index of nilpotency at most 3, i.e. x yzt = 0, for all x, y, z, t ∈ A. We prove that the Itô's theorem for groups remains valid for associative algebras. A structure theorem for metabelian algebras is given in terms of pure linear algebra tools and their classification from the view point of the extension problem is proven. Two border-line cases are worked out in detail: all metabelian algebras having the derived algebra of dimension 1 (resp. codimension 1) are explicitly described and classified. The algebras of the first family are parameterized by bilinear forms and classified by their homothetic relation. The algebras of the second family are parameterized by the set of all matrices (X, Y, u) ∈ M n (k) 2 × k n satisfying X 2 = Y 2 = 0, XY = Y X and Xu = Y u.

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“…The first occurrence of such a study appeared in the setting of Lie algebras in [4]. It is only recently that the associative case has been studied in [1], building on [7]. Yaël Frégier has conjectured that similar theories should exist for most algebraic structures and suggested an approach to unify such a treatment, based on the use of differential graded Lie algebras (dgLa's).…”

Section: Introductionmentioning

confidence: 99%

“…We begin by recalling the results of [1], i.e. the classification of non-abelian extensions in terms of non-abelian cohomology.…”

Section: Introductionmentioning

confidence: 99%

A differential graded Lie algebra approach to non abelian extensions of associative algebras

Gouray

2018

Preprint

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Hochschild products and global non-abelian cohomology for algebras. Applications

Abstract: Abstract. Let A be a unital associative algebra over a field k, E a vector space and π : E → A a surjective linear map with V = Ker(π).

Cited by 5 publications

References 39 publications

On a type of commutative algebras

On a type of commutative algebras

Metabelian Associative Algebras

A differential graded Lie algebra approach to non abelian extensions of associative algebras

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