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1.
Summary. The goal of this paper is to characterise certain probability laws on a class of quantum groups or braided groups that we
will call nilpotent. First we introduce a braided analogue of the Heisenberg–Weyl group, which shall serve as standard example.
We introduce Gaussian functionals on quantum groups or braided groups as functionals that satisfy an analogue of the Bernstein
property, i.e. that the sum and difference of independent random variables are also independent. The corresponding functionals
on the braided line, braided plane and a braided q-Heisenberg–Weyl group are determined. Section 5 deals with continuous convolution
semigroups on nilpotent quantum groups and braided groups. We extend recent results proving the uniqueness of the embedding
of an infinitely divisible probability law into a continuous convolution semigroup for simply connected nilpotent Lie groups
to nilpotent quantum groups and braided groups. Finally, in Section 6 we give some indications how the semigroup approach
of Heyer and Hazod to the Bernstein theorem on groups can be extended to quantum groups and braided groups.
Received: 30 October 1996 / In revised form: 1 April 1997 相似文献
2.
Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut-points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying embeddings of a group into another group, etc. lead to actions of groups on the asymptotic cones, it is natural to consider actions of groups on tree-graded spaces. We develop a theory of such actions which generalizes the well-known theory of groups acting on R-trees. As applications of our theory, we describe, in particular, relatively hyperbolic groups with infinite groups of outer automorphisms, and co-Hopfian relatively hyperbolic groups. 相似文献
3.
Javier Ribón 《Israel Journal of Mathematics》2018,227(1):289-329
We are interested in classifying groups of local biholomorphisms (or even formal diffeomorphisms) that can be endowed with a canonical structure of algebraic groups and their subgroups. Such groups are called finitedimensional. We obtain that cyclic groups, virtually polycyclic groups, finitely generated virtually nilpotent groups and connected Lie groups of local biholomorphisms are finite-dimensional. We provide several methods to identify finite-dimensional groups and build examples.As a consequence we generalize results of Arnold, Seigal–Yakovenko and Binyamini on uniform estimates of local intersection multiplicities to bigger classes of groups, including for example virtually polycyclic groups and in particular finitely generated virtually nilpotent groups. 相似文献
4.
Both the gauge groups and 5-manifolds are important in physics and mathematics. In this paper,we combine them to study the homotopy aspects of gauge groups over 5-manifolds. For principal bundles over non-simply connected oriented closed 5-manifolds of a certain type, we prove various homotopy decompositions of their gauge groups according to different geometric structures on the manifolds, and give the partial solution to the classification of the gauge groups. As applications, we estimate the homotopy exponents of their gauge groups, and show periodicity results of the homotopy groups of gauge groups analogous to the Bott periodicity.Our treatments here are also very effective for rational gauge groups in the general context, and applicable for higher dimensional manifolds. 相似文献
5.
We show that diagram groups can be viewed as fundamental groups of spaces of positive paths on directed 2-complexes (these spaces of paths turn out to be classifying spaces). Thus diagram groups are analogs of second homotopy groups, although diagram groups are as a rule non-Abelian. Part of the paper is a review of the previous results from this point of view. In particular, we show that the so-called rigidity of the R. Thompson's group F and some other groups is similar to the flat torus theorem. We find several finitely presented diagram groups (even of type F∞) each of which contains all countable diagram groups. We show how to compute minimal presentations and homology groups of a large class of diagram groups. We show that the Poincaré series of these groups are rational functions. We prove that all integer homology groups of all diagram groups are free Abelian. 相似文献
6.
Towards a Lie theory of locally convex groups 总被引:3,自引:0,他引:3
Karl-Hermann Neeb 《Japanese Journal of Mathematics》2006,1(2):291-468
In this survey, we report on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modeled
on locally convex spaces, such as integrability of Lie algebras, integrability of Lie subalgebras to Lie subgroups, and integrability
of Lie algebra extensions to Lie group extensions. We further describe how regularity or local exponentiality of a Lie group
can be used to obtain quite satisfactory answers to some of the fundamental problems. These results are illustrated by specialization
to some specific classes of Lie groups, such as direct limit groups, linear Lie groups, groups of smooth maps and groups of
diffeomorphisms. 相似文献
7.
Mohammad Reza R. Moghaddam Ali Reza Salemkar Hossein M. Saanye 《Indagationes Mathematicae》2007,18(1):73-82
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. 相似文献
8.
S. V. Ludkovsky 《Journal of Mathematical Sciences》2008,150(4):2123-2223
Diffeomorphism groups and loop groups of manifolds on Banach spaces over non-Archimedean fields are defined. Moreover, for
these groups, finite-and infinite-dimensional manifolds over the corresponding fields are considered. The group structure,
the differential-geometric structure, and also the topological structure of diffeomorphism groups and loops groups are studied.
We prove that these groups do not locally satisfy the Campbell-Hausdorff formula. The principal distinctions in the structure
for the Archimedean and classical cases are found. The quasi-invariant measures on these groups with respect to dense subgroups
are constructed. Stochastic processes on topological transformation groups of manifolds and, in particular, on diffeomorphism
groups and on loop groups and also the corresponding transition probabilities are constructed. Regular, strongly continuous,
unitary representations of dense subgroups of topological transformation groups of manifolds, in particular, those of diffeomorphism
group and loop groups associated with quasi-invariant measures on groups and also on the corresponding configurational spaces
are constructed. The conditions imposed on the measure and groups under which these unitary representations are irreducible
are found. The induced representations of topological groups are studied by using quasi-invariant measures on topological
groups.
__________
Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions),
Vol. 18, Functional Analysis, 2006. 相似文献
9.
S. V. Ludkovsky 《Journal of Mathematical Sciences》2007,147(3):6703-6846
Diffeomorphism groups and loop groups of manifolds on Banach spaces over non-Archimedean fields are defined. Moreover, for
these groups, finite-and infinite-dimensional manifolds over the corresponding fields are considered. The group structure,
the differential-geometric structure, and also the topological structure of diffeomorphism groups and loops groups are studied.
We prove that these groups do not locally satisfy the Campbell-Hausdorff formula. The principal distinctions in the structure
for the Archimedean and classical cases are found. The quasi-invariant measures on these groups with respect to dense subgroups
are constructed. Stochastic processes on topological transformation groups of manifolds and, in particular, on diffeomorphism
groups and on loop groups and also the corresponding transition probabilities are constructed. Regular, strongly continuous,
unitary representations of dense subgroups of topological transformation groups of manifolds, in particular, those of diffeomorphism
groups and loop groups associated with quasi-invariant measures on groups and also on the corresponding configurational spaces
are constructed. The conditions imposed on the measure and groups under which these unitary representations are irreducible
are found. The induced representations of topological groups are studied by using quasi-invariant measures on topological
groups.
__________
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 39, Functional
Analysis, 2006. 相似文献
10.
For a morphism of quadratic extensions of antistructures, groups similar to the groups of obstructions to splitting along one-sided submanifolds are defined. These groups are a natural generalization of the splitting obstruction groups. The results obtained open additional possibilities for constructing groups and natural maps in L-theory. 相似文献
11.
Peter Eisele 《Journal of Theoretical Probability》1993,6(4):671-691
For a locally compact groupG a condition in terms of probability measures and conjugation is introduced, which implies that limits of shifted convolution powers are always translates of idempotent measures. Such groups are called Tortrat groups. The connection between Tortrat groups and shifted convolution powers is established by the method of tail idempotents. Some construction principles for Tortrat groups are given and applied to show that compact groups, abelian groups, and more generally SIN-groups, as well as MAP-groups and almost connected nilpotent groups are of this type. The class of Tortrat groups is compared with another class investigated by A. Tortrat. 相似文献
12.
We investigate questions related to the minimal degree of invariants of finitely generated diagonalizable groups. These questions were raised in connection to security of a public key cryptosystem based on invariants of diagonalizable groups. We derive results for minimal degrees of invariants of finite groups, abelian groups and algebraic groups. For algebraic groups we relate the minimal degree of the group to the minimal degrees of its tori. Finally, we investigate invariants of certain supergroups that are superanalogs of tori. It is interesting to note that a basis of these invariants is not given by monomials. 相似文献
13.
14.
We give a topological framework for the study of Sela'slimit groups: limit groups are limits of free groups in a compact space of marked groups. Many results get a natural interpretation in
this setting. The class of limit groups is known to coincide with the class of finitely generated fully residually free groups.
The topological approach gives some new insight on the relation between fully residually free groups, the universal theory
of free groups, ultraproducts and non-standard free groups. 相似文献
15.
In this paper, we study Abelian groups that are small with respect to different classes of groups. Completely decomposable
torsion free groups that are small with respect to an arbitrary class of torsion free groups are described completely. Direct
products of groups small with respect to the class of slender groups are derived. 相似文献
16.
The notion of “near isomorphism” for torsion-free Abelian groups of finite rank is well known. In particular, this concept turned out to be of importance for classifying almost completely decomposable groups. We extend near isomorphism to classes of torsion-free Abelian groups of infinite rank which are unions of bcd–groups, this is to say unions of groups which are bounded essential extensions of completely decomposable groups. Moreover, we show that nearly isomorphic groups of this class also have nearly isomorphic endomorphism rings considered as Abelian groups. 相似文献
17.
We describe a class of groups with the property that the finite ones among them are precisely the complex reflection groups of rank 2. This situation is reminiscent of Coxeter groups, among which the finite ones are precisely the real reflection groups. We also study braid relations between complex reflections and indicate connections to an axiomatic study of root systems and to the Shephard–Todd “collineation groups.” 相似文献
18.
Square groups are quadratic analogues of abelian groups. Many properties of abelian groups are shown to hold for square groups. In particular, there is a symmetric monoidal tensor product of square groups generalizing the classical tensor product. 相似文献
19.
Ramesh Prasad Panda 《代数通讯》2018,46(7):3182-3197
In this paper, the minimum degree of power graphs of certain cyclic groups, abelian p-groups, dihedral groups and dicyclic groups are obtained. It is ascertained that the edge-connectivity and minimum degree of power graphs are equal, and consequently, the minimum disconnecting sets of power graphs of the aforementioned groups are determined. In order to investigate the equality of connectivity and minimum degree of power graphs, certain necessary conditions for finite groups and a necessary and su?cient condition for finite cyclic groups are obtained. Moreover, the equality is discussed for the power graphs of abelian p-groups, dihedral groups and dicyclic groups. 相似文献
20.