Group Homomorphism Research Articles

Local cohomology for Gorenstein homologically smooth DG algebras

In this paper, we introduce the theory of local cohomology and local duality to Notherian connected cochain DG algebras. We show that the notion of the local cohomology functor can be used to detect the Gorensteinness of a homologically smooth DG algebra. For any Gorenstein homologically smooth locally finite DG algebra $${\cal A}$$ , we define a group homomorphism $${\rm{Hdet}}:{\rm{Au}}{{\rm{t}}_{dg}}\left( {\cal A} \right) \to {k^ \times }$$ , called the homological determinant. As applications, we present a sufficient condition for the invariant DG subalgebra $${{\cal A}^G}$$ to be Gorenstein, where $${\cal A}$$ is a homologically smooth DG algebra such that $$H\left( {\cal A} \right)$$ is a Noetherian AS-Gorenstein graded algebra and G is a finite subgroup of $${\rm{Au}}{{\rm{t}}_{dg}}\left( {\cal A} \right)$$ . Especially, we can apply this result to DG down-up algebras and non-trivial DG free algebras generated in two degree-one elements.

C0-transport of flux geometry

The goal of this paper is to study, in a large scale point of view, the flux geometry of a closed symplectic manifold (M,ω): namely, the topological counterpart of the flux homomorphism. Using metrics arising from the decomposition of closed 1-forms with respect to an arbitrary linear section S, we generalize the construction of the group of strong symplectic homeomorphisms. The flux homomorphism for symplectomorphisms is extended to a surjective group homomorphism Sω0 on the group of S-homeomorphisms. We prove that the kernel of Sω0 is path connected, coincides with the subgroup Hameo(M,ω) of all Hamiltonian homeomorphisms and investigate the discreteness of the corresponding flux group SΓω. Later on, without appealing to any lifting map, we give an alternative proof of a result from the classical flux geometry saying that any smooth symplectic isotopy in Ham(M,ω) is a Hamiltonian isotopy. Furthermore under some hypothesis, we prove that any S-topological isotopy in Hameo(M,ω) is a continuous Hamiltonian isotopy. We also proved that any S-topological isotopy with trivial flux is homotopic to a continuous Hamiltonian isotopy, relatively to fixed endpoints.

The algebra of the Feistel-Toffoli construction

The process of replacing an arbitrary Boolean function by a bijective one, a fundamental tool in reversible computing and in cryptography, is interpreted algebraically as a particular instance of a certain group homomorphism from the [Formula: see text]-fold cartesian power of a group [Formula: see text] into the automorphism group of the free [Formula: see text]-set over the set [Formula: see text]. It is shown that this construction not only can be generalized from groups to monoids but, more generally, to internal categories in arbitrary finitely complete categories where it becomes a cartesian isomorphism between certain discrete fibrations.

The unrestricted virtual braid groups UV Bn

Let [Formula: see text] and [Formula: see text] be the unrestricted virtual braid group and the unrestricted virtual pure braid group on n strands, respectively. We study the groups [Formula: see text] and [Formula: see text], and our main results are as follows: for [Formula: see text], we give a complete description, up to conjugation, to all possible homomorphisms from [Formula: see text] to the symmetric group [Formula: see text]. For [Formula: see text], we characterize all possible images of [Formula: see text], under a group homomorphism, to any finite group [Formula: see text]. For [Formula: see text], we prove that [Formula: see text] is a characteristic subgroup of [Formula: see text]. In addition, we determine the automorphism group of [Formula: see text] and we prove that [Formula: see text] is a subgroup of the outer automorphism group of [Formula: see text]. Lastly, we show that [Formula: see text] and [Formula: see text] are residually finite and Hopfian but not co-Hopfian. We also remark that some of these results hold accordingly for the welded braid group [Formula: see text] and we discuss about its automorphism group.

A note on Pontryagin duality and continuous logic

We exhibit Pontryagin duality as a special case of Stone duality in a continuous logic setting. More specifically, given an abelian topological group A, and F the family (group) of continuous homomorphisms from A to the circle group T, then, viewing (A,+) equipped with the collection F as a continuous logic structure M, we show that the local type space SF(M) is precisely the Pontryagin dual of the group F where the latter is considered as a discrete group.We conclude, using Pontryagin duality (between compact and discrete abelian groups), that SF(M) is the Bohr compactification of the topological group A.

Abstract group actions of locally compact groups on CAT(0) spaces

We study abstract group actions of locally compact Hausdorff groups on CAT(0) spaces. Under mild assumptions on the action we show that it is continuous or has a global fixed point. This mirrors results by Dudley and Morris–Nickolas for actions on trees. As a consequence we obtain a geometric proof for the fact that any abstract group homomorphism from a locally compact Hausdorff group into a torsion-free CAT(0) group is continuous.

Cataclysms for Anosov representations

In this paper, we construct cataclysm deformations for theta -Anosov representations into a semisimple non-compact connected real Lie group G with finite center, where theta subset Delta is a subset of the simple roots that is invariant under the opposition involution. These generalize Thurston’s cataclysms on Teichmüller space and Dreyer’s cataclysms for Borel-Anosov representations into mathrm {PSL}(n, mathbb {R}). We express the deformation also in terms of the boundary map. Furthermore, we show that cataclysm deformations are additive and behave well with respect to composing a representation with a group homomorphism. Finally, we show that the deformation is injective for Hitchin representations, but not in general for theta -Anosov representations.

Negligible degree two cohomology of finite groups

For a finite group G, a G-module M and a field F, an element u∈Hd(G,M) is negligible over F if for each field extension L/F and every group homomorphism Gal(Lsep/L)→G, u belongs to the kernel of the induced homomorphism Hd(G,M)→Hd(L,M). We determine the group of negligible elements in H2(G,M) for every abelian group M with trivial G-action.

Homomorphisms of Algebraic Groups: Representability and Rigidity

Given two algebraic groups G, H over a field k, we investigate the representability of the functor of morphisms (of schemes) Hom(G,H) and the subfunctor of homomorphisms (of algebraic groups) Homgp(G,H). We show that Hom(G,H) is represented by a group scheme, locally of finite type, if the k-vector space O(G) is finite-dimensional; the converse holds if H is not étale. When G is linearly reductive and H is smooth, we show that Homgp(G,H) is represented by a smooth scheme M; moreover, every orbit of H acting by conjugation on M is open.

Automatic continuity for groups whose torsion subgroups are small

Abstract We prove that a group homomorphism φ : L → G \varphi\colon L\to G from a locally compact Hausdorff group 𝐿 into a discrete group 𝐺 either is continuous, or there exists a normal open subgroup N ⊆ L N\subseteq L such that φ ⁢ ( N ) \varphi(N) is a torsion group provided that 𝐺 does not include ℚ or the 𝑝-adic integers Z p \mathbb{Z}_{p} or the Prüfer 𝑝-group Z ⁢ ( p ∞ ) \mathbb{Z}(p^{\infty}) for any prime 𝑝 as a subgroup, and if the torsion subgroups of 𝐺 are small in the sense that any torsion subgroup of 𝐺 is artinian. In particular, if 𝜑 is surjective and 𝐺 additionally does not have non-trivial normal torsion subgroups, then 𝜑 is continuous. As an application, we obtain results concerning the continuity of group homomorphisms from locally compact Hausdorff groups to many groups from geometric group theory, in particular to automorphism groups of right-angled Artin groups and to Helly groups.

A theorem on zero cycles on surfaces

In this paper we prove a result on 0-cycles on surfaces as an application of the theorem on the kernel of the Gysin homomorphism of Chow groups of 0-cycles of degree zero induced by the embedding of a curve into a surface, and we study the connection of this result with Bloch’s conjecture and constant cycles curves.

Synthetic properties of locally compact groups: preservation and transference

Using techniques from TRO equivalence of masa bimodules we prove various transference results: We show that when $$\alpha $$ is a group homomorphism which pushes forward the Haar measure of G to a measure absolutely continuous with respect to the Haar measure on H, then $$(\alpha \times \alpha )^{-1}$$ preserves sets of compact operator synthesis, and conversely when $$\alpha $$ is onto. We also prove similar preservation results for operator Ditkin sets and operator M-sets, obtaining preservation results for M-sets as corollaries. Some of these results extend or complement existing results of Ludwig, Shulman, Todorov and Turowska.

On q -Rung Orthopair Fuzzy Subgroups

The q -rung orthopair fuzzy environment is an innovative tool to handle uncertain situations in various decision-making problems. In this work, we characterize the idea of a q -rung orthopair fuzzy subgroup and examine various algebraic attributes of this newly defined notion. We also present q -rung orthopair fuzzy coset and q -rung orthopair fuzzy normal subgroup along with relevant fundamental theorems. Moreover, we introduce the concept of q -rung orthopair fuzzy level subgroup and proved related results. At the end, we explore the consequence of group homomorphism on the q -rung orthopair fuzzy subgroup.

On λ-homomorphic skew braces

For a skew left brace ( G , ⋅ , ∘ ) , the map λ : ( G , ∘ ) → Aut ( G , ⋅ ) , a ↦ λ a , where λ a ( b ) = a − 1 ⋅ ( a ∘ b ) for all a , b ∈ G , is a group homomorphism . Then λ can also be viewed as a map from ( G , ⋅ ) to Aut ( G , ⋅ ) , which, in general, may not be a homomorphism. We study skew left braces ( G , ⋅ , ∘ ) for which λ : ( G , ⋅ ) → Aut ( G , ⋅ ) is a homomorphism. Such skew left braces will be called λ -homomorphic. We formulate necessary and sufficient conditions under which a given homomorphism λ : ( G , ⋅ ) → Aut ( G , ⋅ ) gives rise to a skew left brace, which, indeed, is λ -homomorphic. As an application, we construct a lot of skew left braces (of infinite order) on free groups and free abelian groups . We prove that any λ -homomorphic skew left brace is an extension of a trivial skew brace by a trivial skew brace. Special emphasis is given on λ -homomorphic skew left brace for which the image of λ is cyclic. We also obtain set-theoretic solutions of the Yang-Baxter equation corresponding to the skew braces we construct in this paper.

Nilpotent soft polygroups

In this paper, we introduce nilpotent soft (sub)polygroups. In addition, nilpotency of intersection, extended intersection, restricted union of two nilpotent soft polygroups are studied. Espesialy, a necessary and suficient condition between nilpotency of a polygroup and soft polygroups is obtained. Finally, we define two new soft polygroups (Sα)A∪{c} and (Qα)_A derived from a soft polygroup α_A and study on nilpotency of these structures. Also, we extend a soft homomorphism of groups to polygroups. This helps us to extend some properties of groups to polygroups.

Uniformities and a quantale structure on localic groups

Similarly like classical topological groups, the point-free counterparts, localic groups, possess natural uniformities (see e.g. [4,2,10]) obtained from an involutive binary operation on $L$ (roughly corresponding to the algebra of subsets of a classical group). It is the operation that naturally induces the uniformities (even if it would not result from a group) and a study of this aspect of the construction is the main topic of this article. We have here a functor associating with the localic groups quantales of a special type (and with frame group homomorphisms quantale homomorphisms) which are shown to create the uniformities, in fact as a special case of the natural uniformities connected with metric structures. Also, we present a condition under which the quantale allows a reconstruction of the localic group.

Interrelation of nonclassicality conditions through stabiliser group homomorphism

In this paper, we show that coherence witness for a single qubit itself yields conditions for nonlocality and entanglement inequalities for multiqubit systems. It also yields a condition for quantum discord in two-qubit systems. It is shown by employing homomorphism among the stabiliser group of a single qubit and those of multi-qubit states. Interestingly, globally commuting homomorphic images of single-qubit stabilisers do not allow for consistent assignments of outcomes of local observables. We employ these observables for the construction of conditions for nonlocality, entanglement, and quantum discord. As an application, we show that CHSH inequality can be straightforwardly generalised to nonlocality inequalities for multiqubit GHZ states. It also reconfirms the fact that quantumness prevails even in the large N-limit if coherence is sustained. The mapping provides a way to construct many nonlocality inequalities, given a seed inequality. This study gives us a motivation to gain better control over multiple degrees of freedom and multi-party systems. It is because, in multi-party systems, the same nonclassical feature, viz, coherence may appear in many avatars.

Homomorphisms of Lattice-Valued Intuitionistic Fuzzy Subgroup Type-3.

The lattice-valued intuitionistic fuzzy set was introduced by Gerstenkorn and Tepavcevi as a generalization of both the fuzzy set and the L-fuzzy set by incorporating membership functions, nonmembership functions from a nonempty set X to any lattice L, and lattice homomorphism from L to the interval [0,1]. In this article, lattice-valued intuitionistic fuzzy subgroup type-3 (LIFSG-3) is introduced. Lattice-valued intuitionistic fuzzy type-3 normal subgroups, cosets, and quotient groups are defined, and their group theocratic properties are compared with the concepts in classical group theory. LIFSG-3 homomorphism is established and examined in relation to group homomorphism. The research findings are supported by provided examples in each section.

Signature for piecewise continuous groups

Let $\\widehat{\\operatorname{PC}^{\\bowtie}}$ be the group of bijections from $\[0,1\[$ to itself which are continuous outside a finite set. Let $\\operatorname{PC}^{\\bowtie}$ be its quotient by the subgroup of finitely supported permutations. We show that the Kapoudjian class of $\\operatorname{PC}^{\\bowtie}$ vanishes. That is, the quotient map $\\widehat{\\operatorname{PC}^{\\bowtie}}\\rightarrow\\operatorname{PC}^{\\bowtie}$ splits modulo the alternating subgroup of even permutations. This is shown by constructing a nonzero group homomorphism, called signature, from $\\widehat{\\operatorname{PC}^{\\bowtie}}$ to $\\mathbb{Z}/2\\mathbb{Z}$. Then we use this signature to list normal subgroups of every subgroup $\\widehat{G}$ of $\\widehat{\\operatorname{PC}^{\\bowtie}}$ which contains $\\mathfrak{S}\_{\\mathrm{fin}}$ such that $G$, the projection of $\\widehat{G}$ in $\\operatorname{PC}^{\\bowtie}$, is simple.

Higher central charges and Witt groups

In this paper, we introduce the definitions of signatures of braided fusion categories, which are proved to be invariants of their Witt equivalence classes. These signature assignments define group homomorphisms on the Witt group. The higher central charges of pseudounitary modular categories can be expressed in terms of these signatures, which are applied to prove that the Ising modular categories have infinitely many square roots in the Witt group modulo the pointed part. This result is further applied to prove a conjecture of Davydov-Nikshych-Ostrik on the super-Witt group: the torsion subgroup generated by the completely anisotropic s-simple braided fusion categories has infinite rank.