Non-abelian Tensor Product Research Articles

Lie nilpotency and Lie solvability of tensor product of multiplicative Lie algebras

In this article, we discuss Lie nilpotency and Lie solvability of non-abelian tensor product of multiplicative Lie algebras. As our main results, we provide upper bounds on the Lie nilpotency class and Lie solvability length of quotient of tensor product of multiplicative Lie algebras.

On the finiteness of the non-abelian tensor product of groups

In this paper we provide sufficient conditions for the non-abelian tensor product G⊗H to be polycyclic/polycyclic-by-finite in terms of involved groups and derivative subgroups (cf. Theorem 1.1); we also give sufficient conditions for the (local) finiteness of the non-abelian tensor product of groups (cf. Theorems 1.2–1.5). Furthermore, we deduce similar results for some related constructions associated to the non-abelian tensor products, such as the Schur multiplier of a pair of groups M(G,N), the non-abelian q-tensor product M⊗qN, and homotopy pushout (cf. Section 5).

Non-abelian tensor product and circular orderability of groups

For a group G we consider its tensor square G⊗G and exterior square G∧G. We prove that for a circularly orderable group G, under some assumptions on H1(G) and H2(G), its exterior square and tensor square are left-orderable. This yields an obstruction for a circularly orderable group G to have torsion. We apply these results to study circular orderability of tabulated virtual knot groups.

Non-Compatible Action Graph and Its Adjacency Matrix for The Non-abelian Tensor Product for Groups of Prime Power Order

This article focused on the notion of the non-abelian tensor product of groups of prime power order. Particularly, it presented new graph named as Non-compatible action graph and discussed some of its properties. Moreover, this graph concentrated on the case of non-compatible actions of the tensor product of two finite -groups. Furthermore, its adjacency matrix has been determined and discussed in detail. Moreover, the adjacency matrix has been denoted by A(G) and its inputs are 1 whenever there is adjacency and 0 otherwise.

The Adjacency Matrix of The Compatible Action Graph for Finite Cyclic Groups of p-Power Order

Let G and H be two finites -groups, then is the non-abelian tensor product of G and H. In this paper, the compatible action graph for has been considered when and for the two finite -groups by determining the adjacency matrix for and studied some of its properties.

Non-abelian tensor product of precrossed modules in Lie algebras, structure and applications

In the present paper, we introduce the notion of non-abelian tensor product of a given precrossed module by one of its ideals. We use this concept to study the second homology and the nilpotent multiplier of a precrossed module. We also describe the Ganea term for the c-nilpotent multiplier and the universal central extension of precrossed modules in term of the non-abelian tensor product.

Finiteness conditions for the box-tensor product of groups and related constructions

We study finiteness conditions and closure properties for the box-tensor product G⊠H and related constructions. The content of this paper extends some results concerning the non-abelian tensor product of groups G⊗H. In particular, we deduce a quantitative version of the finiteness criterion for the non-abelian tensor product. Moreover, we obtain finiteness conditions for some functors that arise out of the non-abelian tensor square of groups, such as the second homology group H2(G), the non-abelian exterior square G∧G and the second stable homotopy group of an Eilenberg-MacLane space π2S(K(G,1)).

A tensor product approach to compute 2-nilpotent multiplier of p-groups

Let [Formula: see text] be a group given by a free presentation [Formula: see text]. The 2-nilpotent multiplier of [Formula: see text] is the abelian group [Formula: see text] which is invariant of [Formula: see text] [R. Baer, Representations of groups as quotient groups, I, II, and III, Trans. Amer. Math. Soc. 58 (1945) 295–419]. An effective approach to compute the 2-nilpotent multiplier of groups has been proposed by Burns and Ellis [On the nilpotent multipliers of a group, Math. Z. 226 (1997) 405–428], which is based on the nonabelian tensor product. We use this method to determine the explicit structure of [Formula: see text], when [Formula: see text] is a finite (generalized) extra special [Formula: see text]-group. Moreover, the descriptions of the triple tensor product [Formula: see text], and the triple exterior product [Formula: see text] are given.

The non-abelian tensor and exterior products of crossed modules of Lie superalgebras

In this paper, we introduce the notions of non-abelian tensor and exterior products of two ideal graded crossed submodules of a given crossed module of Lie superalgebras. We also study some of their basic properties and their connection with the second homology of crossed modules of Lie superalgebras.

Detecting capable Lie superalgebras

In this article, we show that distributive law holds for non-abelian tensor product of Lie superalgebras under certain direct sums. Thereby we obtain a rule for non-abelian exterior square of a Lie superalgebra. We define capable Lie superalgebra and then give some characterizations. Specifically, we prove that epicenter of a Lie superalgebra is equal to exterior center. Finally, we classify all capable Lie superalgebras whose derived subalgebra dimension is at most one. As an application to those results, we have shown that there exists at least one non-abelian nilpotent capable Lie superalgebra each of dimension (m/n) when

Some structural and closure properties of an extension of the q-tensor product of groups,

In this work we study some structural properties of the group q a non-negative integer, which is an extension of the q-tensor product where G and H are normal subgroups of some group L. We establish by simple arguments some closure properties of when G and H belong to certain Schur classes. This extends similar results concerning the case q = 0 found in the literature. Restricting our considerations to the case G = H, we compute the q-tensor square for q odd, where Dn denotes the dihedral group of order 2n. Upper bounds for the exponent of are also established for nilpotent groups G of class which extend to all similar bounds found by Moravec, P. (2008). The exponents of nonabelian tensor products of groups. J. Pure Appl. Algebra. 212(7):1840–1848.

Universal central extensions and non-abelian tensor product of hom-Lie–Rinehart algebras

In this paper, we study universal central extensions and non-abelian tensor product of hom-Lie–Rinehart algebras. We discuss universal $\alpha $-central extensions, and the lifting of automorphisms and $\alpha $-derivations to central extensions of hom-Li

The non-abelian tensor product of normal crossed submodules of groups

In this article, the notions of non-abelian tensor and exterior products of two normal crossed submodules of a given crossed module of groups are introduced and some of their basic properties are established. In particular, we investigate some common properties between normal crossed modules and their tensor products, and present some bounds on the nilpotency class and solvability length of the tensor product, provided such information is given at least on one of the normal crossed submodules.

Weak commutativity for pro-p groups

We define and study a pro-p version of Sidki’s weak commutativity construction. This is the pro-p group $$\mathfrak {X}_p(G)$$ generated by two copies G and $$G^{\psi }$$ of a pro-p group, subject to the defining relators $$[g,g^{\psi }]$$ for all $$g \in G$$ . We show for instance that if G is finitely presented or analytic pro-p, then $$\mathfrak {X}_p(G)$$ has the same property. Furthermore we study properties of the non-abelian tensor product and the pro-p version of Rocco’s construction $$\nu (H)$$ . We also study finiteness properties of subdirect products of pro-p groups. In particular we prove a pro-p version of the $$(n-1)-n-(n+1)$$ Theorem.

Universal Central Extensions of Internal Crossed Modules via the Non-abelian Tensor Product

In the context of internal crossed modules over a fixed base object in a given semi-abelian category, we use the non-abelian tensor product in order to prove that an object is perfect (in an appropriate sense) if and only if it admits a universal central extension. This extends results of Brown-Loday (in the case of groups) and Edalatzadeh (in the case of Lie algebras). Our aim is to explain how those results can be understood in terms of categorical Galois theory: Edalatzadeh's interpretation in terms of quasi-pointed categories applies, but a more straightforward approach based on the theory developed in a pointed setting by Casas and the second author works as well.

A non-abelian tensor product of precrossed modules in lie algebras

In this paper, we introduce the non-abelian tensor square of precrossed modules in Lie algebras and investigate some of its properties. In particular, for an arbitrary Lie algebra L, we study the relation of the second homology of a precrossed L-module and the non-abelian exterior square. Also, we show how this non-abelian tensor product is related to the universal central extensions (with respect to the subcategory of crossed modules) of a precrossed module.

Subgraph of Compatible Action Graph for Finite Cyclic Groups of p-Power Order

Given two groups G and H, then the nonabelian tensor product G ⊗ H is the group generated by g ⨂ h satisfying the relations gg′ ⊗ h = (gg′ ⊗ gh) (g ⨂ h) and g ⊗ hh′ = (g ⊗ h) (hg′ ⊗ hh) for all g g′ ∈ G and h, h′ ∈ H. If G and H act on each other and each of which acts on itself by conjugation and satisfying (g h)g′ = g(h(g-1 g′)) and (h g)h′ = h(g(h-1 h′)), then the actions are said to be compatible. The action of G on H, gh is a homomorphism from G to a group of automorphism H. If (gh, hg) be a pair of the compatible actions for the nonabelian tensor product of G ⊗ H then ΓG ⊗ H = (V(ΓG ⊗ H), (E(ΓG ⊗ H)) is a compatible action graph with the set of vertices, (V(ΓG ⊗ H) and the set of edges, (E(ΓG ⊗ H). In this paper, the necessary and sufficient conditions for the cyclic subgroups of p-power order acting on each other in a compatible way are given. Hence, a subgraph of a compatible action graph is introduced and its properties are given.

The non-abelian tensor product and the second homology of Leibniz algebras

In the theories of groups and Lie algebras, investigations of the properties of the non-abelian tensor product and their relations to the second homology groups are worthwhile. It is the purpose of the present paper to exhibit such investigations about the non-abelian tensor product of Leibniz algebras. The isomorphism between the non-abelian tensor square and non-abelian exterior square of a Lie algebra L, will enable us to set a simple connection between and . Furthermore, we shall relate the concepts of capability and solvability of a Leibniz algebra to its tensor square. Finally, we give an upper bound for the dimension of the non-abelian tensor square and the second homology of a nilpotent Leibniz algebra in terms of the dimension of its center and derived subalgebra.

Compatible actions of Lie algebras

We study compatible actions (introduced by Brown and Loday in their work on the non-abelian tensor product of groups) in the category of Lie algebras over a fixed ring. We describe the Peiffer product via a new diagrammatic approach, which specializes to the known definitions both in the case of groups and of Lie algebras. We then use this approach to transfer a result linking compatible actions and pairs of crossed modules over a common base object L from groups to Lie algebras. Finally, we show that the Peiffer product, naturally endowed with a crossed module structure, has the universal property of the coproduct in XModL(LieR).

On some properties preserved by the non-abelian tensor product of Hom-Lie algebras

ABSTRACT We study some properties of the non-abelian tensor product of Hom-Lie algebras concerning the preservation of products and quotients, solvability and nilpotency, and describe compatibility with the universal central extensions of perfect Hom-Lie algebras.