Arbitrary Bilinear Form Research Articles

On generalized almost para-Hermitian spaces

Recently, a generalized almost Hermitian metric on an almost complex manifold (M, J) is determined as a generalized Riemannian metric (i.e. an arbitrary bilinear form) G which satisfies G(JX, JY) = G(X,Y), where X and Y are arbitrary vector fields on M. In the same manner we can study a generalized almost para-Hermitian metric and determine almost para-Hermitian spaces. Some properties of these spaces and special generalized almost para-Hermitian spaces including generalized para-Hermitian spaces as well as generalized nearly para-K?hler spaces are determined. Finally, a non-trivial example of generalized almost para-Hermitian space is constructed.

Similarity of quadratic and symmetric bilinear forms in characteristic 2

We say that a field extension L/F has the descent property for isometry (resp. similarity) of quadratic or symmetric bilinear forms if any two forms defined over F that become isometric (resp. similar) over L are already isometric (resp. similar) over F. The famous Artin–Springer theorem states that anisotropic quadratic or symmetric bilinear forms over a field stay anisotropic over an odd degree field extension. As a consequence, odd degree extensions have the descent property for isometry of quadratic as well as symmetric bilinear forms. While this is well known for nonsingular quadratic forms, it is perhaps less well known for arbitrary quadratic or symmetric bilinear forms in characteristic 2. We provide a proof in this situation. More generally, we show that odd degree extensions also have the descent property for similarity. Moreover, for symmetric bilinear forms in characteristic 2, one even has the descent property for isometry and for similarity for arbitrary separable algebraic extensions. We also show Scharlau’s norm principle for arbitrary quadratic or bilinear forms in characteristic 2.

Bilinear covariants and spinor fields duality in quantum Clifford algebras

Classification of quantum spinor fields according to quantum bilinear covariants is introduced in a context of quantum Clifford algebras on Minkowski spacetime. Once the bilinear covariants are expressed in terms of algebraic spinor fields, the duality between spinor and quantum spinor fields can be discussed. Thus, by endowing the underlying spacetime with an arbitrary bilinear form with an antisymmetric part in addition to a symmetric spacetime metric, quantum algebraic spinor fields and deformed bilinear covariants can be constructed. They are thus compared to the classical (non quantum) ones. Classes of quantum spinor fields classes are introduced and compared with Lounesto's spinor field classification. A physical interpretation of the deformed parts and the underlying \documentclass[12pt]{minimal}\begin{document}$\mathbb {Z}$\end{document}Z-grading is proposed. The existence of an arbitrary bilinear form endowing the spacetime already has been explored in the literature in the context of quantum gravity [S. W. Hawking, “The unpredictability of quantum gravity,” Commun. Math. Phys. 87, 395 (1982)]. Here, it is shown further to play a prominent role in the structure of Dirac, Weyl, and Majorana spinor fields, besides the most general flagpoles and flag-dipoles. We introduce a new duality between the standard and the quantum spinor fields, by showing that when Clifford algebras over vector spaces endowed with an arbitrary bilinear form are taken into account, a mixture among the classes does occur. Consequently, novel features regarding the spinor fields can be derived.

Reductivity of the Lie Algebra of a Bilinear Form

Let f: V × V → F be a totally arbitrary bilinear form defined on a finite dimensional vector space V over a field F, and let L(f) be the subalgebra of 𝔤𝔩(V) of all skew-adjoint endomorphisms relative to f. Provided F is algebraically closed of characteristic not 2, we determine all f, up to equivalence, such that L(f) is reductive. As a consequence, we find, over an arbitrary field, necessary and sufficient conditions for L(f) to be simple, semisimple or isomorphic to 𝔰𝔩(n) for some n.

Equivalence and normal forms of bilinear forms

We present an alternative account of the problem of classifying and finding normal forms for arbitrary bilinear forms. Beginning from basic results developed by Riehm, our solution to this problem hinges on the classification of indecomposable forms and in how uniquely they fit together to produce all other forms. We emphasize the use of split forms, i.e., those bilinear forms such that the minimal polynomial of the asymmetry of their non-degenerate part splits over ground field, rather than restricting the field to be algebraically closed. In order to obtain the most explicit results, without resorting to the classification of hermitian, symmetric and quadratic forms, we merely require that the underlying field be quadratically closed.

Error estimates for two-dimensional cross approximation

In this paper we deal with the approximation of a given function f on [ 0 , 1 ] 2 by special bilinear forms ∑ i = 1 k g i ⊗ h i via the so-called cross approximation. In particular we are interested in estimating the error function f − ∑ i = 1 k g i ⊗ h i of the corresponding algorithm in the maximum norm. There is a large amount of publications available that successfully deal with similar matrix algorithms in applied situations, for example in connection with H -matrices (see Boerm and Grasedyck (2003) [9] or Hackbusch (2007) [16] for many references). But as they do not give satisfactory error estimates, we concentrate on the theoretical issues of the problem in the language of functions. We connect it with related results from other areas of analysis in a historical survey and give a lot of references. Our main result is the connection of the error of our algorithm with the error of best approximation by arbitrary bilinear forms. This will be compared with the different approach in Bebendorf (2008) [6].

Computations with Clifford and Grassmann Algebras

Various computations in Grassmann and Clifford algebras can be performed with a Maple package CLIFFORD. It can solve algebraic equations when searching for general elements satisfying certain conditions, solve an eigenvalue problem for a Clifford number, and find its minimal polynomial. It can compute with quaternions, octonions, and matrices with entries in Cl(B) - the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. It uses standard (undotted) Grassmann basis in Cl(Q) but when the antisymmetric part of B is non zero, it can also compute in a dotted Grassmann basis. Some examples of computations are discussed.

Fillmore–Springer–Cnops Construction Implemented in GiNaC

This paper presents an implementation of the Schwerdtfeger-Fillmore-Springer-Cnops construction (SFSCc) along with illustrations of its usage. SFSCc linearises the linear-fraction action of the Moebius group in R^n. This has clear advantages in several theoretical and applied fields including engineering. Our implementation is based on the Clifford algebra capacities of the GiNaC computer algebra system (http://www.ginac.de/), which were described in cs.MS/0410044. The core of this realisation of SFSCc is done for an arbitrary dimension of R^n with a metric given by an arbitrary bilinear form. We also present a subclass for two dimensional cycles (i.e. circles, parabolas and hyperbolas), which add some 2D specific routines including a visualisation to PostScript files through the MetaPost (http://www.tug.org/metapost.html) or Asymptote (http://asymptote.sourceforge.net/) packages. This software is the backbone of many results published in math.CV/0512416 and we use its applications their for demonstration. The library can be ported (with various level of required changes) to other CAS with Clifford algebras capabilities similar to GiNaC. There is an ISO image of a Live Debian DVD attached to this paper as an auxiliary file, a copy is stored on Google Drive as well.

The Hilbert Null-cone on Tuples of Matrices and Bilinear Forms

We describe the null-cone of the representation of G on Mp, where either G = SL(W) × SL(V) and M = Hom(V,W) (linear maps), or G = SL(V) and M is one of the representations S2(V*) (symmetric bilinear forms), Λ2(V*) (skew bilinear forms), or \(V^* \otimes V^*\) (arbitrary bilinear forms). Here V and W are vector spaces over an algebraically closed field K of characteristic zero and Mp is the direct sum of p of copies of M. More specifically, we explicitly determine the irreducible components of the null-cone on Mp. Results of Kraft and Wallach predict that their number stabilises at a certain value of p, and we determine this value. We also answer the question of when the null-cone in Mp is defined by the polarisations of the invariants on M; typically, this is only the case if either dim V or p is small. A fundamental tool in our proofs is the Hilbert–Mumford criterion for nilpotency (also known as unstability).

Mathematics of Clifford - a Maple package for Clifford and Graßmann algebras

CLIFFORD performs various computations in Grassmann and Clifford algebras. It can compute with quaternions, octonions, and matrices with entries in Cl(B) - the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. Two user-selectable algorithms for Clifford product are implemented: 'cmulNUM' - based on Chevalley's recursive formula, and 'cmulRS' - based on non-recursive Rota-Stein sausage. Grassmann and Clifford bases can be used. Properties of reversion in undotted and dotted wedge bases are discussed.

Clifford geometric parameterization of inequivalent vacua

AbstractWe propose a geometric method to parameterize inequivalent vacua by dynamical data. Introducing quantum Clifford algebras with arbitrary bilinear forms we distinguish isomorphic algebras—as Clifford algebras—by different filtrations (resp. induced gradings). The idea of a vacuum is introduced as the unique algebraic projection on the base field embedded in the Clifford algebra, which is however equivalent to the term vacuum in axiomatic quantum field theory and the GNS construction in C*‐algebras. This approach is shown to be equivalent to the usual picture which fixes one product but employs a variety of GNS states. The most striking novelty of the geometric approach is the fact that dynamical data fix uniquely the vacuum and that positivity is not required. The usual concept of a statistical quantum state can be generalized to geometric meaningful but non‐statistical, non‐definite, situations. Furthermore, an algebraization of states takes place. An application to physics is provided by an U (2)‐symmetry producing a gap equation which governs a phase transition. The parameterization of all vacua is explicitly calculated from propagator matrix elements. A discussion of the relation to BCS theory and Bogoliubov–Valatin transformations is given. Copyright © 2001 John Wiley & Sons, Ltd.

Bilinear Forms, Clifford Algebras, q-Commutation Relations, and Quantum Groups

Constructions are described which associate algebras to arbitrary bilinear forms, generalising the usual Clifford and Heisenberg algebras. Quantum groups of symmetries are discussed, both as deformed enveloping algebras and as quantised function spaces. A classification of the equivalence classes of bilinear forms is also given.

Spinor representations of Clifford algebras: a symbolic approach

Clifford algebras Cℓ( B) of an arbitrary, not necessarily symmetric, bilinear form B provide an important computational tool for physicists and an interesting mathematical object to study. In this paper we explain step by step how to compute spinor representations of real Clifford algebras Cℓ( Q) of the quadratic form Q, Q(x) = B(x,x), with a new version of CLIFFORD, a Maple package for computations with Clifford algebras of an arbitrary bilinear form. New procedures in the package follow standard mathematical theory of such representations. When Cℓ( Q) is simple (resp. semisimple) its spinor representation is realized faithfully in a minimal left ideal S = Cℓ( Q) f (resp. S ⊕ S ̂ ) for some primitive idempotent |. The ideal S (resp. S ⊕ S ̂ ) is a right K -module (resp. K ⊕ Ǩ-module) where K is a subalgebra of Cℓ( Q) isomorphic with R, C or H depending on the dimension of V and the signature of Q. We show examples how gamma matrices with entries in K representing basis one-vectors of Cℓ( Q) and scalar products on spinor spaces S for various signatures of Q may be derived with CLIFFORD.

Optimal h-p finite element methods

Abstract A theory of optimal Petrov-Galerkin, h-p version, finite element approximations is presented. The optimal scheme is denned relative to a fine mesh solution space and relative to an arbitrary symmetric bilinear form. The optimal method leads to a symmetric, positive-definite stiffness matrix which is independent of the coefficients of the given problem, exhibits ‘extra superconvergence’ properties, and has a relative error that can be calculated exactly, at each point in the problem domain. Various generalizations are also discussed, including the connection of these methods with certain preconditioning schemes.