Grassmann Structure Research Articles

Geometry on the variety of subspaces with positive definite form: geometry of the Grassmann spaces over generalized hyperbolic spaces

We consider Grassmann structures defined on the family consisting of subspaces on which a given nondegenerate bilinear form defined on a real vector space is positive definite. One may call such structures Grassmann spaces over generalized hyperbolic spaces. We show that the underlying (generalized) hyperbolic space can be recovered in terms of its Grassmannian, and the underlying projective space (equipped with respective associated polarity) can be recovered in terms of the generalized hyperbolic space defined over it.

3D Cadastral Data Model Based on Conformal Geometry Algebra

Three-dimensional (3D) cadastral data models that are based on Euclidean geometry (EG) are incapable of providing a unified representation of geometry and topological relations for 3D spatial units in a cadastral database. This lack of unification causes problems such as complex expression structure and inefficiency in the updating of 3D cadastral objects. The inability of current cadastral data models to express cadastral objects in a unified manner can be attributed to the different expressions of dimensional objects. Because the hierarchical Grassmann structure corresponds to the hierarchical structure of dimensions in conformal geometric algebra (CGA), geometric objects in different dimensions can be constructed by outer products in a unified expression form, which enables the direct extension of two-dimensional (2D) spatial representations to 3D spatial representations. The multivector structure in CGA can be employed to organize and store different dimensional objects in a multidimensional and unified manner. With the advantages of CGA in multidimensional expressions, a new 3D cadastral data model that is based on CGA is proposed in this paper. The geometries and topological relations of 3D spatial units can be represented in a unified form within the multivector structure. Detailed methods for 3D cadastral data model design based on CGA and data organization in CGA are introduced. The new cadastral data model is tested and analyzed with experimental data. The results indicate that the geometry and topological relations of 3D cadastral objects can be represented in a multidimensional manner with an intuitive topological structure and a unified dimensional expression.

Multidimensional-unified topological relations computation: a hierarchical geometric algebra-based approach

This article presents a geometric algebra-based model for topological relation computation. This computational model is composed of three major components: the Grassmann structure preserving hierarchical multivector-tree representation (MVTree), multidimensional unified operators for intersection relation computation, and the judgement rules for assembling the intersections into topological relations. With this model, the intersection relations between the different dimensional objects (nodes at different levels) are computed using the Tree Meet operator. The meet operation between two arbitrary objects is accomplished by transforming the computation into the meet product between each pair of MVTree nodes, which produces a series of intersection relations in the form of MVTree. This intersection tree is then processed through a set of judgement rules to determine the topological relations between two objects in the hierarchy. Case studies of topological relations between two triangles in 3D space are employed to illustrate the model. The results show that with the new model, the topological relations can be computed in a simple way without referring to dimension. This dimensionless way of computing topological relations from geographic data is significant given the increased dimensionality of geographic information in the digital era.

Fermionic Basis in Conformal Field Theory and Thermodynamic Bethe Ansatz for Excited States

We generalize the results of [Comm. Math. Phys. 299 (2010), 825-866, arXiv:0911.3731] (hidden Grassmann structure IV) to the case of excited states of the transfer matrix of the six-vertex model acting in the so-called Matsubara direction. We establish an equivalence between a scaling limit of the partition function of the six-vertex model on a cylinder with quasi-local operators inserted and special boundary conditions, corresponding to particle-hole excitations, on the one hand, and certain three-point correlation functions of conformal field theory (CFT) on the other hand. As in hidden Grassmann structure IV, the fermionic basis developed in previous papers and its conformal limit are used for a description of the quasi-local operators. In paper IV we claimed that in the conformal limit the fermionic creation operators generate a basis equivalent to the basis of the descendant states in the conformal field theory modulo integrals of motion suggested by A. Zamolodchikov (1987). Here we argue that, in order to completely determine the transformation between the above fermionic basis and the basis of descendants in the CFT, we need to involve excitations. On the side of the lattice model we use the excited-state TBA approach. We consider in detail the case of the descendant at level 8.

Hidden Grassmann Structure in the XXZ Model IV: CFT Limit

The Grassmann structure of the critical XXZ spin chain is studied in the limit to conformal field theory. A new description of Virasoro Verma modules is proposed in terms of Zamolodchikov’s integrals of motion and two families of fermionic creation operators. The exact relation to the usual Virasoro description is found up to level 6.

Hidden Grassmann Structure in the XXZ Model II: Creation Operators

In this article we unveil a new structure in the space of operators of the XXZ chain. For each α we consider the space \({\mathcal W_\alpha}\) of all quasi-local operators, which are products of the disorder field \({q^{\alpha \sum_{j=-\infty}^0\sigma ^3_j}}\) with arbitrary local operators. In analogy with CFT the disorder operator itself is considered as primary field. In our previous paper, we have introduced the annhilation operators b(ζ), c(ζ) which mutually anti-commute and kill the “primary field”. Here we construct the creation counterpart b*(ζ), c*(ζ) and prove the canonical anti-commutation relations with the annihilation operators. We conjecture that the creation operators mutually anti-commute, thereby upgrading the Grassmann structure to the fermionic structure. The bosonic operator t*(ζ) is the generating function of the adjoint action by local integrals of motion, and commutes entirely with the fermionic creation and annihilation operators. Operators b*(ζ), c*(ζ), t*(ζ) create quasi-local operators starting from the primary field. We show that the ground state averages of quasi-local operators created in this way are given by determinants.

Hidden Grassmann Structure in the XXZ Model

For the critical XXZ model, we consider the space $$\mathcal{W}_{[\alpha]}$$ of operators which are products of local operators with a disorder operator. We introduce two anti-commutative families of operators $${\bf {b}}(\zeta), {\bf {c}}(\zeta)$$ which act on $$\mathcal{W}_{[\alpha]}$$ . These operators are constructed as traces over representations of the q-oscillator algebra, in close analogy with Baxter’s Q-operators. We show that the vacuum expectation values of operators in $$\mathcal{W}_{[\alpha]}$$ can be expressed in terms of an exponential of a quadratic form of $${\bf {b}}(\zeta), {\bf {c}}(\zeta)$$ .

CURVATURE SUBMODULES FOR GRASSMANN STRUCTURES WITH TORSION

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.

Geometric Quantities of Manifolds with grassmann structure

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.

Yang–Mills connections over manifolds with Grassmann structure

Let M be a manifold with Grassmann structure, i.e., with an isomorphism of the cotangent bundle T*M≅E⊗H with the tensor product of two vector bundles E and H. We define the notion of a half-flat connection ∇W in a vector bundle W→M as a connection whose curvature F∈S2E⊗Λ2H⊗End W ⊂ Λ2T*M⊗End W. Under appropriate assumptions, for example, when the Grassmann structure is associated with a quaternionic Kähler structure on M, half-flatness implies the Yang–Mills equations. Inspired by the harmonic space approach, we develop a local construction of (holomorphic) half-flat connections ∇W over a complex manifold with (holomorphic) Grassmann structure equipped with a suitable linear connection. Any such connection ∇W can be obtained from a prepotential by solving a system of linear first order ODEs. The construction can be applied, for instance, to the complexification of hyper-Kähler manifolds or more generally to hyper-Kähler manifolds with admissible torsion and to their higher-spin analogs. It yields solutions of the Yang–Mills equations.

Grassmann structures and integrable systems

We consider some natural connections which arise between right-flat ( p, q) paraconformal structures and integrable systems. We find that such systems may be formulated in Lax form with a “Lax p-tuple” of linear differential operators, depending a spectral parameter which lives in ( q−1)-dimensional complex projective space. Generally, the differential operators contain partial derivatives with respect to the spectral parameter.

Semiintegrable almost Grassmann structures

In the present paper we study locally semiflat (we also call them semiintegrable) almost Grassmann structures. We establish necessary and sufficient conditions for an almost Grassmann structure to be α- or β-semiintegrable. These conditions are expressed in terms of the fundamental tensors of almost Grassmann structures. Since we are not able to prove the existence of locally semiflat almost Grassmann structures in the general case, we give many examples of α- and β-semiintegrable structures, mostly four-dimensional. For all examples we find systems of differential equations of the families of integral submanifolds V α and V β of the distributions Δ α and Δ β of plane elements associated with an almost Grassmann structure. For some examples we were able to integrate these systems and find closed form equations of submanifolds V α and V β .

Conformal and Grassmann structures

The main results on the theory of conformal and almost Grassmann structures are presented. The common properties of these structures and also the differences between them are outlined. In particular, the structure groups of these structures and their differential prolongations are found. A complete system of geometric objects of the almost Grassmann structure totally defining its geometric structure is determined. The vanishing of these objects determines a locally Grassmann manifold. It is proved that the integrable almost Grassmann structures are locally Grassmann. The criteria of semiintegrability of almost Grassmann structures is proved in invariant form.

Twistorial transformation of conformally einstein quaternionic manifolds

Let M be the complexification of the 4k-dimensional (k ~ 2) real analytic quaternionic manifold [i]. M has the Grassmann structure TM = S ~ S (where TM is the holomorphic tangent bundle and S and g are locally free holomorphic bundles of ranks 2k and 2, respectively), which, together with the identification A2ks = A2~, can be interpreted as the generalization of the four-dimensional conformal structure to the case of 4k dimensions [2]. The locally free bundle of local twistors T of rank 2k + 2 is defined by the following exact sequence:

Mesopic color matching: some theoretical issues

In this paper I address some theoretical issues concerning the additivity laws of Grassmann as they might apply to color matches in mesopic vision (in which light intensities are such that both rods and cones are active). It is first shown that rods and cones have linearly independent spectral responses, and hence rod and cone spectral responses span a four-dimensional matching space. Next, the iterative tetrachromatic-matching experiment of Trezona [Vision Res. 13, 9 (1973)] is examined for consistency with a four-dimensional Grassmann structure; convergence conditions for the technique are derived and shown to be consistent with the actual conditions of the experiment. Finally, a discussion is presented of the conditions under which mesopic color matching would require only three primaries, even when color matching obeys four-dimensional (but not three-dimensional) Grassmann laws. A two-dimensional rod-cone diagram gives significant but not complete insight into the conditions for trichromacy in a four-dimensional color-matching space.

On the local structure of the Euclidean Dirac field

A simple Green’s formula for the Euclidean Dirac operator in Schwinger’s real formalism allows us to study some localization properties of the Euclidean Dirac field. A rather complete analysis follows of the connection between the Grassmann structure of the Euclidean field and the Clifford structure of the field at sharp time, analyzed in terms of its independent degrees of freedom.

Color measurement and color theory: I. Representation theorem for Grassmann structures

For trichromatic color measurement, the empirically based structure consists of the set of colored lights, with its operations of additive mixture and scalar multiplication, and the binary relation of metameric matching. The representing numerical structure is a vector space. The important axioms are Grassmann's laws. The vector representation is constructed in a canonical or coordinate-free manner, mainly using Grassmann's additivity law. Trichromacy is used only to fix the dimensionality. Color theories attempt to get a more unique homomorphism by enriching the basic empirical structure with new empirical relations, subject to new axioms. Examples of such enriching relations include: discriminability or dissimilarity ordering of color pairs; dichromatic matching relations; and unidimensional matching relations, or codes. Representation theorems for the latter two examples are based on Grassmann-type laws also. The relationship between a Grassmann structure and its unidimensional Grassmann codes is modeled by the relationship between a vector space and its dual space of linear functionals. Dual spaces are used to clarify theorems relating to the three-pigment hypothesis and to reduction dichromacy.