Classifying complements for associative algebras

Agore, A. L.

doi:10.1016/j.laa.2014.01.012

Cited by 14 publications

(16 citation statements)

References 15 publications

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“…In Section 3 we provide some explicit examples. Let S n be the symmetric group and C n the cyclic group of order n. By applying our results to the factorization S n = S n−1 C n we obtain the following: (1) any group H of order n is isomorphic to (C n ) r , the r-deformation of the cyclic group C n for some deformation map r : C n → S n−1 of the canonical matched pair (S n−1 , C n , ⊲, ⊳) and (2) the number of isomorphism types of all groups of order n is equal to | D(C n , S n−1 | (⊲, ⊳)) |. Therefore, we obtain a combinatorial formula for computing the number of isomorphism types of all groups of order n which arises from a minimal set of data: the factorization S n = S n−1 C n .…”

Section: Introductionmentioning

confidence: 86%

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Classifying complements for groups. Applications

Agore¹,

Militaru²

2015

Ann. inst. Fourier

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Let A ≤ G be a subgroup of a group G. An A-complement of G is a subgroup H of G such that G = AH and A ∩ H = {1}. The classifying complements problem asks for the description and classification of all A-complements of G. We shall give the answer to this problem in three steps. Let H be a given A-complement of G and (⊲, ⊳) the canonical left/right actions associated to the factorization G = AH. To start with, H is deformed to a new A-complement of G, denoted by Hr, using a certain map r : H → A called a deformation map of the matched pair (A, H, ⊲, ⊳). Then the description of all complements is given: H is an A-complement of G if and only if H is isomorphic to Hr, for some deformation map r : H → A. Finally, the classification of complements proves that there exists a bijection between the isomorphism classes of all A-complements of G and a cohomological object D (H, A | (⊲, ⊳)). As an application we show that the theoretical formula for computing the number of isomorphism types of all groups of order n arises only from the factorization Sn = Sn−1Cn.

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Section: Introductionmentioning

confidence: 86%

“…The results presented here for groups can be used as a model for developing similar theories in the fields listed above. For Hopf algebras and Lie algebras we refer to [4] and for associative algebras to [1].…”

Section: Introductionmentioning

confidence: 99%

Classifying complements for groups. Applications

Agore¹,

Militaru²

2015

Ann. inst. Fourier

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“…From the point of view of algebra, this problem is natural and significant. Similar problems for classical algebraic objects such as associative algebras, Hopf algebras, Lie algebras, Leibniz algebras and left-symmetric algebras have been investigated in [1,2,3,10]. According to the theory of bicrossed product developed in [12] and [10], it is easy to see that E in the classifying complements problem is just a bicrossed product of R and Q associated to a matched pair.…”

Section: Introductionmentioning

confidence: 99%

Classifying complements for conformal algebras

Hong

2020

Preprint

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Let R ⊆ E be two Lie conformal algebras and Q be a given complement of R in E. Classifying complements problem asks for describing and classifying all complements of R in E up to an isomorphism. It is known that E is isomorphic to a bicrossed product of R and Q. We show that any complement of R in E is isomorphic to a deformation of Q associated to the bicrossed product. A classifying object is constructed to parameterize all R-complements of E. Several explicit examples are provided. Similarly, we also develop a classifying complements theory of associative conformal algebras.

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“…At the level of Lie, or more Communicated by Miin Huey Ang. B G. Militaru gigel.militaru@gmail.com; gigel.militaru@fmi.unibuc.ro 1 Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei 14, 010014 Bucharest 1, Romania general Leibniz algebras, the corresponding concept of metabelian Lie/Leibniz algebra, as 2-step solvable algebra, is also well known [2,6,7].…”

Section: Introductionmentioning

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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Classifying complements for associative algebras

Cited by 14 publications

References 15 publications

Classifying complements for groups. Applications

Classifying complements for groups. Applications

Classifying complements for conformal algebras

Metabelian Associative Algebras

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