Separability criteria via wavelet transform on homogenous spaces and projective representations

Open Physics ◽  
2010 ◽  
Vol 8 (3) ◽  
Author(s):  
Mahdi Karamati ◽  
Mohammad Rezapour
Keyword(s):  
Group Theory ◽  
Quantum Entanglement ◽  
Projective Representation ◽  
Permutation Groups ◽  
Characteristic Functions ◽  
Compact Groups ◽  
Reconstruction Method ◽  
Separability Criteria ◽  
Homogenous Space ◽  
Projective Representations

AbstractThe intimate connection between the Banach space wavelet reconstruction method for each unitary representation of a given group and homogenous space, and the quantum entanglement description using group theory were both studied in our previous articles. Here, we present a universal description of quantum entanglement using group theory and non-commutative characteristic functions for homogenous space and projective representation of compact groups on Banach spaces for some well known examples, such as: Moyal representation for a spin; Dihedral and Permutation groups.

scholarly journals Projective representations of quivers

2002 ◽  
Vol 31 (2) ◽  
pp. 97-101 ◽  
Author(s):  
Sangwon Park
Keyword(s):  
Projective Representation ◽  
Representations Of Quivers ◽  
Direct Sum ◽  
Projective Representations

We prove thatP1 →f P2is a projective representation of a quiverQ=•→•if and only ifP1andP2are projective leftR-modules,fis an injection, andf (P 1)⊂P 2is a summand. Then, we generalize the result so that a representationM1 →f1  M2  →f2⋯→fn−2  Mn−1→fn−1  Mnof a quiverQ=•→•→•⋯•→•→•is projective representation if and only if eachMiis a projective leftR-module and the representation is a direct sum of projective representations.

Elements of Group Theory

2021 ◽  
pp. 51-110
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Group Theory ◽  
Lie Groups ◽  
Lorentz Group ◽  
Unitary Representations ◽  
Compact Groups ◽  
Mathematical Language ◽  
Group Representations ◽  
Infinite Groups ◽  
Infinite Dimensional ◽  
Symmetry Properties

The mathematical language which encodes the symmetry properties in physics is group theory. In this chapter we recall the main results. We introduce the concepts of finite and infinite groups, that of group representations and the Clebsch–Gordan decomposition. We study, in particular, Lie groups and Lie algebras and give the Cartan classification. Some simple examples include the groups U(1), SU(2) – and its connection to O(3) – and SU(3). We use the method of Young tableaux in order to find the properties of products of irreducible representations. Among the non-compact groups we focus on the Lorentz group, its relation with O(4) and SL(2,C), and its representations. We construct the space of physical states using the infinite-dimensional unitary representations of the Poincaré group.

Holomorphic and abstract inducing

1984 ◽  
Vol 96 (3) ◽  
pp. 453-468 ◽  
Author(s):  
K. C. Hannabuss
Keyword(s):  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Spaces ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Induced Representations ◽  
Projective Representations ◽  
Kernel Hilbert Spaces ◽  
Cohomology Spaces

AbstractA method of constructing projective representations of separable locally compact groups in reproducing kernel Hilbert spaces is presented, based on the generalized inducing process of Rieffel and Fell. Examples show that the method can be used to construct some well-known holomorphically induced representations. Some representations on cohomology spaces are also described.

GROUP THEORETICAL APPROACH TO QUANTUM ENTANGLEMENT AND TOMOGRAPHY WITH WAVELET TRANSFORM IN BANACH SPACES

2007 ◽  
Vol 05 (03) ◽  
pp. 367-386 ◽  
Author(s):  
M. REZAEE ◽  
M. A. JAFARIZADEH ◽  
M. MIRZAEE
Keyword(s):  
Banach Space ◽  
Wavelet Transform ◽  
Banach Spaces ◽  
Theoretical Approach ◽  
Unitary Group ◽  
Atomic Decomposition ◽  
Rotation Group ◽  
Reconstruction Method ◽  
Banach Frame ◽  
Separability Criteria

The intimate connection between the Banach space wavelet reconstruction method for each unitary representation of a given group and some of well-known quantum tomographies, such as tomography of rotation group, spinor tomography and tomography of unitary group, is established. Also both the atomic decomposition and Banach frame nature of these quantum tomographic examples are revealed in detail. Finally, we consider separability criteria for any state with group theoretical wavelet transform on Banach spaces.

On the Smallest Degrees of Projective Representations of the Groups PSL(n, q)

1971 ◽  
Vol 23 (1) ◽  
pp. 90-102 ◽  
Author(s):  
Morton E. Harris ◽  
Christoph Hering
Keyword(s):  
Lower Bound ◽  
Higher Order ◽  
Projective Representation ◽  
Minimal Degree ◽  
Prime Integer ◽  
Order Of Magnitude ◽  
Projective Representations

In this paper, we obtain information about the minimal degree δ of any non-trivial projective representation of the group PSL(n, q) with n ≧ 2 over an arbitrary given field K. Our main results for the groups PSL(n, q) (Theorems 4.2, 4.3, and 4.4) state that, apart from certain exceptional cases with small n, we have the following rather surprising situation: if q = pf (where p is a prime integer) and char K = p, then δ = n, but if q = pf and char K ≠ p, then δ is of a considerably higher order of magnitude, namely, δ is at least qn–l – 1 or if n = 2 and q is odd. Note that for n = 2, this lower bound for δ is the best possible. However, for n ≧ 3, this lower bound can conceivably be improved.

scholarly journals Compact groups with many elements of bounded order

2020 ◽  
Vol 23 (6) ◽  
pp. 991-998
Author(s):  
Meisam Soleimani Malekan ◽  
Alireza Abdollahi ◽  
Mahdi Ebrahimi
Keyword(s):  
Group Theory ◽  
Haar Measure ◽  
Open Subgroup ◽  
Profinite Group ◽  
Compact Groups ◽  
Profinite Groups ◽  
Russian Academy Of Sciences ◽  
Kourovka Notebook ◽  
Commuting Pairs ◽  
Academy Of Sciences

AbstractLévai and Pyber proposed the following as a conjecture: Let G be a profinite group such that the set of solutions of the equation {x^{n}=1} has positive Haar measure. Then G has an open subgroup H and an element t such that all elements of the coset tH have order dividing n (see [V. D. Mazurov and E. I. Khukhro, Unsolved Problems in Group Theory. The Kourovka Notebook. No. 19, Russian Academy of Sciences, Novosibirisk, 2019; Problem 14.53]). The validity of the conjecture has been proved in [L. Lévai and L. Pyber, Profinite groups with many commuting pairs or involutions, Arch. Math. (Basel) 75 2000, 1–7] for {n=2}. Here we study the conjecture for compact groups G which are not necessarily profinite and {n=3}; we show that in the latter case the group G contains an open normal 2-Engel subgroup.

Projective Representations of Minimum Degree of Group Extensions

1978 ◽  
Vol 30 (5) ◽  
pp. 1092-1102 ◽  
Author(s):  
Walter Feit ◽  
Jacques Tits
Keyword(s):  
Simple Group ◽  
Central Extension ◽  
Finite Simple Group ◽  
Minimum Degree ◽  
Projective Representation ◽  
Closed Field ◽  
Natural Question ◽  
Central Extensions ◽  
Ordinary Representation ◽  
Projective Representations

Let G be a finite simple group and let F be an algebraically closed field. A faithful projective F-representation of G of smallest possible degree often cannot be lifted to an ordinary representation of G, though it can of course be lifted to an ordinary representation of some central extension of G. It is a natural question to ask whether by considering non-central extensions, it is possible in some cases to decrease the smallest degree of a faithful projective representation.

scholarly journals Covering group theory for locally compact groups

2001 ◽  
Vol 114 (2) ◽  
pp. 187-199 ◽  
Author(s):  
Valera Berestovskii ◽  
Conrad Plaut
Keyword(s):  
Group Theory ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Covering Group

scholarly journals Groups whose Projective character degrees are powers of a prime

1988 ◽  
Vol 30 (2) ◽  
pp. 177-180 ◽  
Author(s):  
R. J. Higgs
Keyword(s):  
Finite Group ◽  
Projective Representation ◽  
Character Degrees ◽  
Projective Character ◽  
Projective Representations ◽  
A Finite Group

Let Gbe a finite group, and P:G → GL(n, ) be such that for all x, y ∈ G(i) P(x)P(y) = α(x, y)P(xy), and(ii)P(l) = In,where α(x, y) ∈ *; then P is a projective representation of G with cocycle α and degree n. For other basic definitions concerning projective representations see [4].

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