scholarly journals The non-abelian tensor product of groups and related constructions

1989 ◽  
Vol 31 (1) ◽  
pp. 17-29 ◽  
Author(s):  
N. D. Gilbert ◽  
P. J. Higgins
Keyword(s):  
Tensor Product ◽  
Special Cases ◽  
Algebraic Description ◽  
Close Relationship ◽  
Closed Structure ◽  
Product Of Groups

The tensor product of two arbitrary groups acting on each other was introduced by R. Brown and J.-L. Loday in [5, 6]. It arose from consideration of the pushout of crossed squares in connection with applications of a van Kampen theorem for crossed squares. Special cases of the product had previously been studied by A. S.-T. Lue [10] and R. K. Dennis [7]. The tensor product of crossed complexes was introduced by R. Brown and the second author [3] in connection with the fundamental crossed complex π(X) of a filtered space X, which also satisfies a van Kampen theorem. This tensor product provides an algebraic description of the crossed complex π(X ⊗ Y) and gives a symmetric monoidal closed structure to the category of crossed complexes (over groupoids). Both constructions involve non-abelian bilinearity conditions which are versions of standard identities between group commutators. Since any group can be viewed as a crossed complex of rank 1, a close relationship might be expected between the two products. One purpose of this paper is to display the direct connections that exist between them and to clarify their differences.

Extension of normal functionals on W*-tensor products

1975 ◽  
Vol 78 (2) ◽  
pp. 301-307 ◽  
Author(s):  
Simon Wassermann
Keyword(s):  
Representation Theory ◽  
Tensor Product ◽  
Hilbert Spaces ◽  
General Result ◽  
Von Neumann Algebras ◽  
Tensor Products ◽  
Von Neumann ◽  
Hilbert Algebras ◽  
Special Cases

A deep result in the theory of W*-tensor products, the Commutation theorem, states that if M and N are W*-algebras faithfully represented as von Neumann algebras on the Hilbert spaces H and K, respectively, then the commutant in L(H ⊗ K) of the W*-tensor product of M and N coincides with the W*-tensor product of M′ and N′. Although special cases of this theorem were established successively by Misonou (2) and Sakai (3), the validity of the general result remained conjectural until the advent of the Tomita-Takesaki theory of Modular Hilbert algebras (6). As formulated, the Commutation theorem is a spatial result; that is, the W*-algebras in its statement are taken to act on specific Hilbert spaces. Not surprisingly, therefore, known proofs rely heavily on techniques of representation theory.

On Nilpotent Products of Cyclic Groups

1960 ◽  
Vol 12 ◽  
pp. 447-462 ◽  
Author(s):  
Ruth Rebekka Struik
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Finite Abelian Group ◽  
Nilpotent Groups ◽  
Multiplication Table ◽  
Direct Products ◽  
Free Products ◽  
Cyclic Groups ◽  
Special Cases ◽  
Product Of Groups

In this paper G = F/Fn is studied for F a free product of a finite number of cyclic groups, and Fn the normal subgroup generated by commutators of weight n. The case of n = 4 is completely treated (F/F2 is well known; F/F3 is completely treated in (2)); special cases of n > 4 are studied; a partial conjecture is offered in regard to the unsolved cases. For n = 4 a multiplication table and other properties are given.The problem arose from Golovin's work on nilpotent products ((1), (2), (3)) which are of interest because they are generalizations of the free and direct product of groups: all nilpotent groups are factor groups of nilpotent products in the same sense that all groups are factor groups of free products, and all Abelian groups are factor groups of direct products. In particular (as is well known) every finite Abelian group is a direct product of cyclic groups. Hence it becomes of interest to investigate nilpotent products of finite cyclic groups.

scholarly journals LIFTING THEOREMS FOR TENSOR FUNCTORS ON MODULE CATEGORIES

2011 ◽  
Vol 10 (01) ◽  
pp. 129-155 ◽  
Author(s):  
ROBERT WISBAUER
Keyword(s):  
General Theory ◽  
Tensor Product ◽  
Category Theory ◽  
Systematic Approach ◽  
Wreath Products ◽  
Baxter Equation ◽  
General Category ◽  
Module Theory ◽  
Elementary Module ◽  
Special Cases

Any (co)ring R is an endofunctor with (co)multiplication on the category of abelian groups. These notions were generalized to monads and comonads on arbitrary categories. Starting around 1970 with papers by Beck, Barr and others a rich theory of the interplay between such endofunctors was elaborated based on distributive laws between them and Applegate's lifting theorem of functors between categories to related (co)module categories. Curiously enough some of these results were not noticed by researchers in module theory and thus notions like entwining structures and smash products between algebras and coalgebras were introduced (in the nineties) without being aware that these are special cases of the more general theory. The purpose of this survey is to explain several of these notions and recent results from general category theory in the language of elementary module theory focusing on functors between module categories given by tensoring with a bimodule. This provides a simple and systematic approach to smash products, wreath products, corings and rings over corings (C-rings). We also highlight the relevance of the Yang–Baxter equation for the structures on the threefold tensor product of algebras or coalgebras (see 3.6).

The Krull and global dimension of the tensor product of quantum tori

2016 ◽  
Vol 15 (09) ◽  
pp. 1650174
Author(s):  
Ashish Gupta
Keyword(s):  
Tensor Product ◽  
Upper Bound ◽  
Tensor Products ◽  
Global Dimension ◽  
Base Field ◽  
Quantum Torus ◽  
Common Value ◽  
Quantum Tori ◽  
Special Cases ◽  
The Common

An [Formula: see text]-dimensional quantum torus is defined as the [Formula: see text]-algebra generated by variables [Formula: see text] together with their inverses satisfying the relations [Formula: see text], where [Formula: see text]. The Krull and global dimensions of this algebra are known to coincide and the common value is equal to the supremum of the rank of certain subgroups of [Formula: see text] that can be associated with this algebra. In this paper we study how these dimensions behave with respect to taking tensor products of quantum tori over the base field. We derive a best possible upper bound for the dimension of such a tensor product and from this special cases in which the dimension is additive with respect to tensoring.

Solvency II, Undertaking Specific Parameters (USPs) Validation, Generalization and Criticism

2021 ◽  
Author(s):  
Alexandros Α. Zimbidis
Keyword(s):  
Solvency Ii ◽  
System Response ◽  
Mathematical Framework ◽  
Insurance Company ◽  
Gamma Distributions ◽  
Special Cases ◽  
Input Signals ◽  
Close Relationship ◽  
Legislative Framework

This paper provides a deeper actuarial insight in the mathematics and algorithms described in Delegate Act 35/2015 of Solvency II legislative framework with respect to the determination of Undertaking Specific Parameters (USPs). The numerical investigation is based on typical input signals used by control engineers in order to check the system response. It is finally revealed the close relationship between the USPs values and the values of the typical function of standard deviation. Finally, a generalization of the mathematical framework covering the special cases of the Pareto and Gamma distributions as inputs for the aggregate losses of an insurance company is provided.

scholarly journals Subdivision Depth Computation for Tensor Productn-Ary Volumetric Models

2011 ◽  
Vol 2011 ◽  
pp. 1-22 ◽  
Author(s):  
Ghulam Mustafa ◽  
Muhammad Sadiq Hashmi
Keyword(s):  
Tensor Product ◽  
Error Bounds ◽  
Error Control ◽  
Computational Formula ◽  
Evaluation Technique ◽  
Volumetric Models ◽  
Higher Dimensional ◽  
Special Cases ◽  
Subdivision Depth ◽  
Control Tool

We offer computational formula of subdivision depth for tensor productn-ary (n⩾2) volumetric models based on error bound evaluation technique. This formula provides and error control tool in subdivision schemes over regular hexahedron lattice in higher-dimensional spaces. Moreover, the error bounds of Mustafa et al. (2006) are special cases of our bounds.

Non-abelian tensor and exterior products of multiplicative Lie rings

2017 ◽  
Vol 29 (3) ◽  
Author(s):  
Guram Donadze ◽  
Nick Inassaridze ◽  
Manuel Ladra
Keyword(s):  
Tensor Product ◽  
Lie Rings ◽  
Product Of Groups

AbstractWe define a non-abelian tensor product of multiplicative Lie rings. This is a new concept providing a common approach to the non-abelian tensor product of groups defined by Brown and Loday and to the non-abelian tensor product of Lie rings defined by Ellis. We also prove an analogue of Miller’s theorem for multiplicative Lie rings.

scholarly journals Finiteness conditions for the non-abelian tensor product of groups

2017 ◽  
Vol 187 (4) ◽  
pp. 603-615 ◽  
Author(s):  
R. Bastos ◽  
I. N. Nakaoka ◽  
N. R. Rocco
Keyword(s):  
Tensor Product ◽  
Finiteness Conditions ◽  
Product Of Groups

scholarly journals Shift Unitary Transform for Constructing Two-Dimensional Wavelet Filters

2011 ◽  
Vol 2011 ◽  
pp. 1-19
Author(s):  
Fei Li ◽  
Jianwei Yang
Keyword(s):  
Tensor Product ◽  
Perfect Reconstruction ◽  
Two Dimensional ◽  
Orthogonal Wavelet ◽  
One Dimensional ◽  
Wavelet Filters ◽  
Parametrization Method ◽  
Special Cases ◽  
Unitary Transform ◽  
Quadrature Filter

Due to the difficulty for constructing two-dimensional wavelet filters, the commonly used wavelet filters are tensor-product of one-dimensional wavelet filters. In some applications, more perfect reconstruction filters should be provided. In this paper, we introduce a transformation which is referred to asShift Unitary Transform(SUT) ofConjugate Quadrature Filter(CQF). In terms of this transformation, we propose a parametrization method for constructing two-dimensional orthogonal wavelet filters. It is proved that tensor-product wavelet filters are only special cases of this parametrization method. To show this, we introduce the SUT of one-dimensional CQF and present a complete parametrization of one-dimensional wavelet system. As a result, more ways are provided to randomly generate two-dimensional perfect reconstruction filters.

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