scholarly journals Geometric Quantities of Manifolds with grassmann structure

2005 ◽  
Vol 180 ◽  
pp. 45-76 ◽  
Author(s):  
N. Bokan ◽  
P. Matzeu ◽  
Z. Rakić
Keyword(s):  
Tensor Product ◽  
Point Of View ◽  
Vector Spaces ◽  
Grassmann Manifolds ◽  
Simple Modules ◽  
Complete Decomposition ◽  
Curvature Tensors ◽  
Torsion Free ◽  
Product Vector

AbstractWe study geometry of manifolds endowed with a Grassmann structure which depends on symmetries of their curvature. Due to this reason a complete decomposition of the space of curvature tensors over tensor product vector spaces into simple modules under the action of the group G = GL(p, ℝ) ⊗ GL(q, ℝ) is given. The dimensions of the simple submodules, the highest weights and some projections are determined. New torsion-free connections on Grassmann manifolds apart from previously known examples are given. We use algebraic results to reveal obstructions to the existence of corresponding connections compatible with some type of normalizations and to enlighten previously known results from another point of view.

CURVATURE SUBMODULES FOR GRASSMANN STRUCTURES WITH TORSION

2006 ◽  
Vol 03 (05n06) ◽  
pp. 975-993
Author(s):  
N. BOKAN ◽  
Z. RAKIĆ
Keyword(s):  
Representation Theory ◽  
Tensor Product ◽  
Grassmann Manifold ◽  
Point Of View ◽  
Vector Spaces ◽  
Grassmann Manifolds ◽  
Flat Connections ◽  
Complete Decomposition ◽  
Theory Point ◽  
Curvature Tensors

A complete decomposition of the space [Formula: see text] of the curvature tensors over tensor product of vector spaces into simple modules under the action of the group G = GL(p, ℝ) ⊗ GL(q, ℝ) is given. We use these results to study geometry of manifolds with Grassmann structure and Grassmann manifolds endowed with a connection whose torsion is not zero. We show that Oscr M a manifold is an example of a manifold with Grassmann structure. Owing to this fact, we consider results of Miron, Atanasiu, Anastasiei, Čomić and others from representation theory point of view and connect them with some results of Alekseevsky, Cortes, and Devchand, as well as of Machida and Sato, and others. New examples of connections with torsion defined on four-dimensional Grassmann manifold are given. Symmetries of curvatures for half-flat connections are also investigated. We use algebraic results to reveal obstructions to the existence of corresponding connections.

The Torsion Free Pieri Formula

1998 ◽  
Vol 50 (2) ◽  
pp. 266-289 ◽  
Author(s):  
D. J. Britten ◽  
F. W. Lemire
Keyword(s):  
Tensor Product ◽  
Arbitrary Degree ◽  
Simple Modules ◽  
Infinite Dimensional ◽  
Complete Reducibility ◽  
Finite Dimensional ◽  
Pieri Formula ◽  
Free Modules ◽  
Central Characters ◽  
Torsion Free

AbstractCentral to the study of simple infinite dimensional g𝓵(n, C)-modules having finite dimensional weight spaces are the torsion free modules. All degree 1 torsion free modules are known. Torsion free modules of arbitrary degree can be constructed by tensoring torsion free modules of degree 1 with finite dimensional simple modules. In this paper, the central characters of such a tensor product module are shown to be given by a Pieri-like formula, complete reducibility is established when these central characters are distinct and an example is presented illustrating the existence of a nonsimple indecomposable submodule when these characters are not distinct.

scholarly journals Tensor Products and Bimorphisms

1976 ◽  
Vol 19 (4) ◽  
pp. 385-402 ◽  
Author(s):  
Bernhard Banaschewski ◽  
Evelyn Nelson
Keyword(s):  
Tensor Product ◽  
Commutative Ring ◽  
Tensor Products ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Bilinear Maps ◽  
Special Situation ◽  
Dual Spaces ◽  
Finite Dimensional

The binary tensor product, for modules over a commutative ring, has two different aspects: its connection with universal bilinear maps and its adjointness to the internal hom-functor. Furthermore, in the special situation of finite-dimensional vector spaces, the tensor product can also be described in terms of dual spaces and the internal hom-functor. The aim of this paper is to investigate these relationships in the setting of arbitrary concrete categories.

A class of simple non-weight modules over the virasoro algebra

2020 ◽  
Vol 63 (4) ◽  
pp. 956-970 ◽  
Author(s):  
Haibo Chen ◽  
JianZhi Han
Keyword(s):  
Tensor Product ◽  
Sufficient Conditions ◽  
Virasoro Algebra ◽  
Vector Spaces ◽  
Complex Numbers ◽  
Infinite Dimensional ◽  
Highest Weight ◽  
Alpha Omega ◽  
Necessary And Sufficient ◽  
Infinite Dimensional Lie Algebra

AbstractThe Virasoro algebra $\mathcal {L}$ is an infinite-dimensional Lie algebra with basis {Lm, C| m ∈ ℤ} and relations [Lm, Ln] = (n − m)Lm+n + δm+n,0((m3 − m)/12)C, [Lm, C] = 0 for m, n ∈ ℤ. Let $\mathfrak a$ be the subalgebra of $\mathcal {L}$ spanned by Li for i ≥ −1. For any triple (μ, λ, α) of complex numbers with μ ≠ 0, λ ≠ 0 and any non-trivial $\mathfrak a$-module V satisfying the condition: for any v ∈ V there exists a non-negative integer m such that Liv = 0 for all i ≥ m, non-weight $\mathcal {L}$-modules on the linear tensor product of V and ℂ[∂], denoted by $\mathcal {M}(V,\mu ,\Omega (\lambda ,\alpha ))\ (\Omega (\lambda ,\alpha )=\mathbb {C}[\partial ]$ as vector spaces), are constructed in this paper. We prove that $\mathcal {M}(V,\mu ,\Omega (\lambda ,\alpha ))$ is simple if and only if μ ≠ 1, λ ≠ 0, α ≠ 0. We also give necessary and sufficient conditions for two such simple $\mathcal {L}$-modules being isomorphic. Finally, these simple $\mathcal {L}$-modules $\mathcal {M}(V,\mu ,\Omega (\lambda ,\alpha ))$ are proved to be new for V not being the highest weight $\mathfrak a$-module whose highest weight is non-zero.

scholarly journals The Bloch-Vector Space for N-Level Systems: the Spherical-Coordinate Point of View

2005 ◽  
Vol 12 (03) ◽  
pp. 207-229 ◽  
Author(s):  
Gen Kimura ◽  
Andrzej Kossakowski
Keyword(s):  
Vector Space ◽  
Quantum State ◽  
Point Of View ◽  
Vector Spaces ◽  
Bloch Vector ◽  
State Spaces ◽  
Spherical Coordinate ◽  
N Level ◽  
Dual Property

Bloch-vector spaces for N-level systems are investigated from the spherical-coordinate point of view in order to understand their geometrical aspects. We present a characterization of the space by using the spectra of (orthogonal) generators of SU (N). As an application, we find a dual property of the space which provides an overall picture of the space. We also provide three classes of quantum-state representations based on actual measurements and discuss their state-spaces.

scholarly journals A COMPLETELY ENTANGLED SUBSPACE OF MAXIMAL DIMENSION

2006 ◽  
Vol 04 (02) ◽  
pp. 325-330 ◽  
Author(s):  
B. V. RAJARAMA BHAT
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Product Formula ◽  
Maximal Dimension ◽  
Maximum Dimension ◽  
Dimension Formula ◽  
Finite Dimensional ◽  
Minimum Dimension ◽  
Orthogonal Product ◽  
Product Vector

Consider a tensor product [Formula: see text] of finite-dimensional Hilbert spaces with dimension [Formula: see text], 1 ≤ i ≤ k. Then the maximum dimension possible for a subspace of [Formula: see text] with no non-zero product vector is known to be d1 d2…dk - (d1 + d2 + … + dk + k - 1. We obtain an explicit example of a subspace of this kind. We determine the set of product vectors in its orthogonal complement and show that it has the minimum dimension possible for an unextendible product basis of not necessarily orthogonal product vectors.

scholarly journals The relation of spatial and tensor product of Arveson systems — the random set point of view

2015 ◽  
Vol 18 (04) ◽  
pp. 1550029
Author(s):  
Volkmar Liebscher
Keyword(s):  
Tensor Product ◽  
Important Implication ◽  
Point Of View ◽  
Random Set ◽  
Bessel Processes ◽  
Set Point ◽  
Zero Sets ◽  
Continuous Range

We characterise the embedding of the spatial product of two Arveson systems into their tensor product using the random set technique. An important implication is that the spatial tensor product does not depend on the choice of the reference units, i.e. it is an intrinsic construction. There is a continuous range of examples coming from the zero sets of Bessel processes where the two products do not coincide. The lattice of all subsystems of the tensor product is analysed in different cases. As a by-product, the Arveson systems coming from Bessel zeros prove to be primitive in the sense of Ref. 12.

scholarly journals ON THE ASPHERICITY OF LENGTH-6 RELATIVE PRESENTATIONS WITH TORSION-FREE COEFFICIENTS

2008 ◽  
Vol 51 (1) ◽  
pp. 201-214
Author(s):  
Seong Kun Kim
Keyword(s):  
Free Group ◽  
Zero Divisor ◽  
Point Of View ◽  
Torsion Free Group ◽  
Torsion Free

AbstractAn interesting result of Ivanov implies that a non-aspherical relative presentation that defines a torsion-free group would provide a potential counterexample to the Kaplansky zero-divisor conjecture. In this point of view, we prove the asphericity of the length-6 relative presentation $\langle H,x: xh_1xh_2xh_3xh_4xh_5xh_6\rangle$, provided that each coefficient is torsion free.

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