Semidirect products in ideally exact categoriesd

This paper aims at providing a systematic treatment of the crystallographic point groups. Some well-known properties of them, in terms of the theory of the poles of finite rotations, are first discussed, so as to provide a simple way for recognizing their invariant subgroups. A definition of the semi-direct product is then given, and it is shown that all crystallographic point groups can be expressed as a semi-direct product of one of their invariant subgroups by a cyclic subgroup. Many useful relations between point groups can be obtained by exploiting the properties of the triple and mixed triple semi-direct products, which are defined. Much of the rest of the paper is devoted to the theory of the representations of semi-direct products. The treatment here parallels that given by Seitz (1936) for the reduction of space groups in terms of the representations of its invariant subgroups (the translation groups). The latter, however, are always Abelian and this is not always the case for point groups. The full treatment of the general case, such as given by McIntosh (1958), is laborious and it is shown that, if the emphasis is placed on the bases of the representations, rather than the representations themselves, it is possible to achieve the reduction of the point groups by a method hardly more involved than that required when the invariant subgroup is Abelian. It is also shown that, just as for space groups, the representations of the invariant subgroups can be denoted and visualized by means of a vector, which allows a very rapid classification of the representations, very much as the k vector as used by Bouckaert, Smoluchowski & Wigner (1936) allows the formalism of the Seitz method for space groups to be carried out in a graphical fashion. One of the major consequences of this work is that it affords a substantial simplification in the use of the symmetrizing and projection operators that are required to obtain symmetry-adapted functions: a very systematic alternative to the method given by Melvin (1956) is therefore provided. In the last section of the paper all the techniques discussed are applied in detail, as an example, to the cubic groups. The projection operators are used to obtain symmetry-adapted spherical harmonics for these groups. The paper might be found useful as an introduction to the methods for the reduction of space groups.

Semidirect products of categories and applications

A discrete group which admits a faithful, finite dimensional, linear representation over a field F of characteristic zero is called linear.This note combines the natural structure of semi-direct products with work of A. Lubotzky [13] on the existence of linear representations to develop a technique to give sufficient conditions to show that a semi-direct product is linear.Let G denote a discrete group which is a semi-direct product given by a split extensionThis note defines an additional type of structure for this semi-direct product called a stable extension below.The main results are as follows:(1) If π and Γ are linear, and the extension is stable, then G is also linear.Restrictions concerning this extension are necessary to guarantee that G is linear as seen from properties of the Formanek-Procesi "poison group" [7].(2) If the action of Γ on π has a "Galois-like" property that it factors through the automorphisms of certain natural "towers of groups over π" ( to be defined below ), then the associated extension is stable and thus G is linear.(3) The condition of a stable extension also implies that G admits filtration quotients which themselves give a natural structure of Lie algebra and

Let G = A ⋊ B be a semi-direct product of two groups. We investigate conditions under which the subgroup of Aut(G) fixing A as a set is also a semi-direct product. If A is characteristic in G, then we have conditions under which Aut(G) itself is a semi-direct product.

The aim of this paper is to study \(\lambda \)-semidirect and \(\lambda \)-Zappa-Szep products of restriction semigroups. The former concept was introduced for inverse semigroups by Billhardt, and has been extended to some classes of left restriction semigroups. The latter was introduced, again in the inverse case, by Gilbert and Wazzan. We unify these concepts by considering what we name the scaffold of a Zappa-Szep product \(S\bowtie T\) where S and T are restriction. Under certain conditions this scaffold becomes a category. If one action is trivial, or if S is a semilattice and T a monoid, the scaffold may be ordered so that it becomes an inductive category. A standard technique, developed by Lawson and based on the Ehresmann-Schein-Nambooripad result for inverse semigroups, allows us to define a product on our category. We thus obtain restriction semigroups that are \(\lambda \)-semidirect products and \(\lambda \)-Zappa-Szep products, extending the work of Billhardt and of Gilbert and Wazzan. Finally, we explicate the internal structure of \(\lambda \)-semidirect products.

Semidirect products and crossed modules in monoids with operations

In this paper we describe the new model of the representations of the current groups with a semisimple Lie group of the rank one. In the earlier papers of 70-80-th (Araki, Gelfand-Graev-Vershik) had posed the problem about irreducible representations of the current group for $SL(2,R)$, and was used for this the well-known Fock space-structure That construction could be applied to the arbitrary locally compact group,and is based on a so called special representation of the original group $G$, with nontrivial 1-cohomology. A new construction uses the special property of one dimensional extensions (semi-direct product)of the nilpotent groups which allows immediately to produce the special representation of the group and then to apply the quasi-Poisson construction from the previous papers by authors in order to construct the representation of current group. The parabolic subgroup of the semisimple Lie group of rank one has such semidirect product, and special representation of it can be extended onto whole semisimple group. As result we obtain needed new model of the irreducible representation of semi-simple current groups.

It is known that the moduli space of smooth Fano–Mukai fourfolds $$V_{18}$$ of genus 10 has dimension one. We show that any such fourfold is a completion of $${\mathbb {C}}^4$$ in two different ways. Up to isomorphism, there is a unique fourfold $$V_{18}^{{\mathrm {s}}}$$ acted upon by $${\mathrm{SL}}_2({\mathbb {C}})$$ . The group is a semidirect product . Furthermore, $$V_{18}^{{\mathrm {s}}}$$ is a $${\mathrm{GL}}_2({\mathbb {C}})$$ -equivariant completion of $${\mathbb {C}}^4$$ , and as well of $${\mathrm{GL}}_2({\mathbb {C}})$$ . The restriction of the $${\mathrm{GL}}_2({\mathbb {C}})$$ -action on $$V_{18}^{{\mathrm {s}}}$$ to yields a faithful representation with an open orbit. There is also a unique, up to isomorphism, fourfold $$V_{18}^{\mathrm a}$$ such that the group is a semidirect product . For a Fano–Mukai fourfold $$V_{18}$$ isomorphic neither to $$V_{18}^{{\mathrm {s}}}$$ , nor to $$V_{18}^{\mathrm a}$$ , the group is a semidirect product of $$({{\mathbb {G}}}_{\mathrm {m}})^2$$ and a finite cyclic group whose order is a factor of 6. Besides, we establish that the affine cone over any polarized Fano–Mukai variety $$V_{18}$$ is flexible in codimension one, and flexible if $$V_{18}=V_{18}^{{\mathrm {s}}}$$ .

We establish analogs of the three Bieberbach theorems for a lattice \(\Gamma\) in a semidirect product \(\mathsf{K}\rtimes\mathsf{K}\) where \(\mathsf{S}\) is a connected, simply connected solvable Lie group and \(\mathsf{K}\) is a compact subgroup of its automorphism group. We first prove that the action of \(\Gamma\) on \(\mathsf{S}\) is metrically equivalent to an action of \(\Gamma\) on a supersolvable Lie group. The latter is shown to be determined by \(\Gamma\) itself up to an affine diffeomorphism. Then we characterize these lattices algebraically as polycrystallographic groups. Furthermore, we realize any polycrystallographic group \(\Gamma\) as a lattice in a semidirect product \(\mathsf{S}\rtimes\mathsf{F}\) with \(\mathsf{F}\) being a finite group whose order is bounded by a constant only depending on the dimension of \(\mathsf{S}\). This generalization of the first Bieberbach theorem is used to obtain a partial generalization of the third one as well. Finally we show for any torsion free closed subgroup \(\Upsilon \subset \mathsf{K}\rtimes\mathsf{K}\) that the quotient \(\mathsf{S}/\Upsilon\) is the total space of a vector bundle over a compact manifold B, where B is the quotient of a solvable Lie group by a torsion free polycrystallographic group.

The first part of this chapter treats Lie algebras, beginning with definitions and many examples. The notions of solvable, nilpotent, radical, semisimple, and simple are introduced, and these notions are followed by a discussion of the effect of a change of the underlying field. The idea of a semidirect product begins the development of the main structural theorems for real Lie algebras—the iterated construction of all solvable Lie algebras from derivations and semidirect products, Lie's Theorem for solvable Lie algebras, Engel's Theorem in connection with nilpotent Lie algebras, and Cartan's criteria for solvability and semisimplicity in terms of the Killing form. From Cartan's Criterion for Semisimplicity, it follows that semisimple Lie algebras are direct sums of simple Lie algebras. Cartan's Criterion for Semisimplicity is used also to provide a long list of classical examples of semisimple Lie algebras. Some of these examples are defined in terms of quaternion matrices. Quaternion matrices of size n-by-n may be related to complex matrices of size 2n-by-2n. The treatment of Lie algebras concludes with a study of the finite-dimensional complex-linear representations of sl[(2, ℂ). There is a classification theorem for the irreducible representations of this kind, and the general representations are direct sums of irreducible ones. Section 10 contains a review of the elementary theory of Lie groups and their Lie algebras. The abstract theory as in Chevalley [1946] is summarized, and the correspondence is made with the concrete theory of closed linear groups, where the Lie algebra is obtained as the space of derivatives at t = 0 of smooth curves in the group passing through the identity at t = 0. The section ends with a discussion of the adjoint representation. The remainder of the chapter explores some aspects of the connection between Lie groups and Lie algebras. One aspect is the relationship between automorphisms and derivations. The derivations of a semisimple Lie algebra are inner, and consequently the identity component of the group of automorphisms of a semisimple Lie algebra consists of inner automorphisms. In addition, simply connected solvable Lie groups may be built one dimension at a time as semidirect products with ℝ1, and consequently they are diffeomorphic to Euclidean space. For simply connected nilpotent groups the exponential map is itself a diffeomorphism. The earlier long list of classical semisimple Lie algebras corresponds to a list of the classical semisimple Lie groups. The issue that needs attention for these groups is their connectedness, and this is proved by using the polar decomposition of matrices.

We study an effective gauge theory whose gauge group is a semidirect product G = G c ⋊ Γ with G c and Γ being a connected Lie group and a finite group, respectively. The semidirect product is defined through a projective homomorphism (i.e. homomorphism up to the center of G c ) from Γ into G c . To be specific, we take SU (3) L as G c and ℤ3 × ℤ3 as Γ. We notice that the irreducible representations of the gauge group G necessarily contain three G c -multiplets in spite of the Abelian nature of Γ = ℤ3 × ℤ3. This triplication phenomenon is due to the semidirect product structure of G. We suggest that the appearance of three families is attributable to this triplication. We give a toy model on the lepton mixing and show that under a particular vacuum alignment the tri-bimaximal mixing matrix is reproduced.

Rees Matrix Covers and Semidirect Products of Regular Semigroups

In the quest in constructing conformal field theories (CFTs), Jones has discovered a beautiful and deep connection between CFT, Richard Thompson’s groups, and knot theory. This led to a powerful functorial framework for constructing actions of particular groups arising from categories such as Thompson’s groups and braid groups. In particular, given a group and two of its endomorphisms one can construct a semidirect product where the largest Thompson’s group $V$ is acting. These semidirect products have remarkable diagrammatic descriptions that were previously used to provide new examples of groups having the Haagerup property. They naturally appear in certain field theories as being generated by local and global symmetries. Moreover, these groups occur in a construction of Tanushevski and can be realised using Brin–Zappa–Szep’s products with the technology of cloning systems of Witzel and Zaremsky. We consider in this article the class of groups obtained in that way where one of the endomorphism is trivial leaving the case of two nontrivial endomorphisms to a 2nd article. We provide an explicit description of all these groups as permutational restricted twisted wreath products where $V$ is the group acting and the twist depends on the endomorphism chosen. We classify this class of groups up to isomorphisms and provide a thin description of their automorphism group thanks to an unexpected rigidity phenomena.

For commutative Banach algebras B and I and a continuous representation π: B → M(I), where M(I) are those bounded linear operators T: I → I that satisfy T(ij) = iT(j) for all i,jeI, the direct sum B I can be made into a commutative Banach algebra which contains B and I as respectively a closed subalgebra and a closed ideal. This algebra is called the semidirect product of B and I. Some topological aspects of the character space of a semidirect product are described. Furthermore, decompositions of commutative Banach algebras into the direct sum of a subalgebra and a principal ideal are investigated.

A generalized Baumslag-Solitar group (GBS group) is a finitely generated group $G$ which acts on a tree with all edge and vertex stabilizers infinite cyclic. We show that Out(G) either contains non-abelian free groups or is virtually nilpotent of class at most 2. It has torsion only at finitely many primes. One may decide algorithmically whether Out(G) is virtually nilpotent or not. If it is, one may decide whether it is virtually abelian, or finitely generated. The isomorphism problem is solvable among GBS groups with Out(G) virtually nilpotent. If $G$ is unimodular (virtually $F_n \times Z$), then Out(G) is commensurable with a semi-direct product $Z^k \rtimes Out(H)$ with $H$ virtually free.