Representations of certain semidirect product groups

We study a class of semidirect product groups G = N · U where N is a generalized Heisenberg group and U is a generalized indefinite unitary group. This class contains the Poincaré group and the parabolic subgroups of the simple Lie groups of real rank 1. The unitary representations of G and (in the unimodular cases) the Plancherel formula for G are written out. The problem of computing Mackey obstructions is completely avoided by realizing the Fock representations of N on certain U-invariant holomorphic cohomology spaces.     

左中心rpp半群的半直积

讨论了幂等元都是左中心元的rpp半群的半直积,给出了这种半群半直积的充要条件,推广了一些已知的结果.     

A bijection between the adjoint and coadjoint orbits of a semidirect product

We prove that there exists a geometric bijection between the sets of adjoint and coadjoint orbits of a semidirect product, provided a similar bijection holds for particular subgroups. We also show that under certain conditions the homotopy types of any two orbits in bijection with each other are the same. We apply our theory to the examples of the affine group and the Poincaré group, and discuss the limitations and extent of this result to other groups.     

Equations on the semidirect product of a finite semilattice by a finite commutative monoid

LetCom _t,q denote the variety of finite monoids that satisfy the equationsxy=yx andx ^t =x ^t+q . The varietyCom _1,1 is the variety of finite semilattices also denoted byJ ₁. In this paper, we consider the product varietyJ ₁*Com _t,q generated by all semidirect products of the formM*N withM∈J ₁ andN∈Com _t,q . We give a complete sequence of equations forJ ₁*Com _t,q implying complete sequences of equations forJ ₁*(Com∩A),J ₁*(Com∩G) andJ ₁*Com, whereCom denotes the variety of finite commutative monoids,A the variety of finite aperiodic monoids andG the variety of finite groups. This material is based upon work supported by the National Science Foundation under Grants No. CCR-9101800 and CCR-9300738. Many thanks to the referee for his valuable comments and suggestions.     

The Baer-invariant and the direct limit

In this note we establish that the Baer-invariant of a group commutes with direct limit. It has a very useful application so that if one works with the Baerinvariant, or in particular Schur-multiplicator, of arbitrary groups it is only needed to take the groups to be finitely generated.     

Weak geodesic flow on a semidirect product and global solutions to the periodic Hunter-Saxton system

We give explicit solutions for the two-component Hunter-Saxton system on the unit circle. Moreover, we show how global weak solutions can be naturally constructed using the geometric interpretation of this system as a re-expression of the geodesic flow on the semidirect product of a suitable subgroup of the diffeomorphism group of the circle with the space of smooth functions on the circle. These spatially and temporally periodic solutions turn out to be conservative.     

Calculation of the Baer-invariant of certain groups

The present paper is devoted to calculation of the Baer-invariants of the dihedral and the quaternion groups of order 8, and also a certain class ofp-groups with respect to the variety of nilpotent groups.     

Some inequalities for the Baer-invariant of a pair of finite groups

In this paper we introduce the concept of Baer-invariant of a pair of groups with respect to a variety of groups v. Some inequalities for the Baer-invariant of a pair of finite groups are obtained, when v is considered to be the Schur-Baer variety. We also present a condition for which the order of the Baerinvariant of a pair of finite groups divides the order of the Baer-invariant of their factor groups. Finally, some inequalities for the Schur-multiplier of a pair of finite nilpotent groups and their factor groups are given.     

Subgroups of semidirect products

The notion of a reduced crossed homomorphism is introduced and subgroups of a semidirect product are described by means of it. Subsemidirect products and semidirect products with a given structure of normal subgroups are characterized.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 1048–1055, July–August, 1991.     

On the genus of the semidirect product of ℤ9 by ℤ3

We show that the non-commutative semidirect product Γ of ?₉ by ?₃ has orientable genus 4. In other words, some Cayley graph of Γ embeds in an orientable surface of genus 4 (Euler characteristic ?6), but no Cayley graph of Γ embeds in an orientable surface of genus less than 4 (Euler characteristic greater than ?6). We also show that some Cayley graph of Γ embeds in a (non-orientable) surface of Euler characteristic ?3, but no Cayley graph of Γ embeds in a surface of Euler characteristic greater than ?3. Γ is the first known example of a group whose orientable Euler characteristic and non-orientable Euler characteristic differ by more than 1. Our results also complete the determination of the orientable genus of each group of order less than 32.     

Hecke algebras of semidirect products

We consider group-subgroup pairs in which the group is a semidirect product and the subgroup is contained in the normal part. We give conditions for the pair to be a Hecke pair and we show that the enveloping Hecke algebra and Hecke -algebra are canonically isomorphic to semigroup crossed products, generalizing earlier results of Arledge, Laca and Raeburn and of Brenken.

     

Automorphism groups of semidirect products

This paper shows that if is a semidirect product of finite groups, then if and only if and for all . As an application, we investigate the automorphism group of a split metacyclic p-group for odd p. The second author is supported by the Natural Science Foundation of China (10671058).     

The semidirect products of finite cyclic groups that areI-E groups

AnI-E group is a group in which the endomorphism near-ring generated by the group's inner automorphisms equals the endomorphism near-ring generated by its endomorphisms. In this paper we shall completely determine the finite groups that are semidirect products of cyclic groups and areI-E groups.     

A method for approximation of the exponential map in semidirect product of matrix Lie groups and some applications

In this paper we explore the computation of the matrix exponential in a manner that is consistent with Lie group structure. Our point of departure is the decomposition of Lie algebra as the semidirect product of two Lie subspaces and an application of the Baker-Campbell-Hausdorff formula. Our results extend the results in Iserles and Zanna (2005) [2], Zanna and Munthe-Kaas(2001/02) [4] to a range of Lie groups: the Lie group of all solid motions in Euclidean space, the Lorentz Lie group of all solid motions in Minkowski space and the group of all invertible (upper) triangular matrices. In our method, the matrix exponential group can be computed by a less computational cost and is more accurate than the current methods. In addition, by this method the approximated matrix exponential belongs to the corresponding Lie group.     

Weyl quantization for semidirect products

Let G be the semidirect product V?K where K is a connected semisimple non-compact Lie group acting linearily on a finite-dimensional real vector space V. Let O be a coadjoint orbit of G associated by the Kirillov-Kostant method of orbits with a unitary irreducible representation π of G. We consider the case when the corresponding little group K₀ is a maximal compact subgroup of K. We realize the representation π on a Hilbert space of functions on Rn where n=dim(K)−dim(K₀). By dequantizing π we then construct a symplectomorphism between the orbit O and the product R2n×O^′ where O^′ is a little group coadjoint orbit. This allows us to obtain a Weyl correspondence on O which is adapted to the representation π in the sense of [B. Cahen, Quantification d'une orbite massive d'un groupe de Poincaré généralisé, C. R. Acad. Sci. Paris Série I 325 (1997) 803-806]. In particular we recover well-known results for the Poincaré group.