Commutative Field Research Articles

Symmetric Minkowski planes ordered by separation

In Karzel et al. (J. Geom. 99: 116–127, 2009) we introduced for a symmetric Minkowski plane \({ {\mathfrak M} := (P,\Lambda,{\mathfrak G}_1,{\mathfrak G}_2) }\) an order concept by the notion of an orthogonal valuation for the circles of Λ and showed that there is a one to one correspondence between the valuations and the halforderings of the accompanying commutative field. Here we consider an order concept which is based on the notion of separation for quadruples of concyclic points and establish the connections between these two notions. Our main result (cf. Theorem 3.3) states that these concepts are equivalent.

On the energy crisis in noncommutative [formula omitted] model

We study the CP ( 1 ) system in ( 2 + 1 ) -dimensional noncommutative space with and without Chern–Simons term. Using the Seiberg–Witten map we convert the noncommutative CP ( 1 ) system to an action written in terms of the commutative fields. We find that this system presents the same infinite size instanton solution as the commutative Chern–Simons- CP ( 1 ) model without a potential term. Based on this result we argue that the BPS equations are compatible with the full variational equations of motion, rejecting the hypothesis of an “energy crisis”. In addition we examine the noncommutative CP ( 1 ) system with a Chern–Simons interaction. In this case we find that when the theory is transformed by the Seiberg–Witten map it also presents the same instanton solution as the commutative Chern–Simons- CP ( 1 ) model.

Special Moufang sets with abelian Hua subgroup

In this paper we complete the classification of special Moufang sets M ( U , τ ) with abelian Hua subgroup. It is already known that if U has odd characteristic, then M ( U , τ ) is the projective line over a commutative field. We present a similar result which also deals with special Moufang sets of even characteristic.

Cauchon diagrams for quantized enveloping algebras

Let g be a finite-dimensional complex simple Lie algebra, K a commutative field and q a nonzero element of K which is not a root of unity. To each reduced decomposition of the longest element w 0 of the Weyl group W corresponds a PBW basis of the quantised enveloping algebra U q + ( g ) , and one can apply the theory of deleting-derivation to this iterated Ore extension. In particular, for each decomposition of w 0 , this theory constructs a bijection between the set of prime ideals in U q + ( g ) that are invariant under a natural torus action and certain combinatorial objects called Cauchon diagrams. In this paper, we give an algorithmic description of these Cauchon diagrams when the chosen reduced decomposition of w 0 corresponds to a good ordering (in the sense of Lusztig (1990) [Lus90]) of the set of positive roots. This algorithmic description is based on the constraints that are coming from Lusztig's admissible planes Lusztig (1990) [Lus90]: each admissible plane leads to a set of constraints that a diagram has to satisfy to be Cauchon. Moreover, we explicitly describe the set of Cauchon diagrams for explicit reduced decomposition of w 0 in each possible type. In any case, we check that the number of Cauchon diagrams is always equal to the cardinal of W. In Cauchon and Mériaux (2008) [CM08], we use these results to prove that Cauchon diagrams correspond canonically to the positive subexpressions of w 0 . So the results of this paper also give an algorithmic description of the positive subexpressions of any reduced decomposition of w 0 corresponding to a good ordering.

Quelques endomorphismes continus des suites P-récursives

Let be K a commutative field of zero characteristic, r ( K ) and j ( K ) the Hadamard algebra of linear recurrence sequences with constant coefficients respectively polynomial coefficients. In a preceding article, we have defined the map decimation ϕ d and tressage ψ d , J , σ , where d ∈ N * , J ∈ N d and σ is a permutation of { 0 , 1 , … , d − 1 } . We have shown that these maps are continuous endomorphisms of the algebra r ( K ) . In this Note, we show that these maps are also continuous endomorphisms of the algebra j ( K ) . To cite this article: A. Ait Mokhtar, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Power series over generalized Krull domains

We resolve an open problem in commutative algebra and Field Arithmetic, posed by Jarden. Let R be a generalized Krull domain. Is the ring R 〚 X 〛 of formal power series over R a generalized Krull domain? We show that the answer is negative. Moreover, we show that essentially the opposite theorem holds. We prove that if R is a generalized Krull domain which is not a Krull domain, then R 〚 X 〛 is never a generalized Krull domain.

GENERALIZATIONS OF THE FUNDAMENTAL THEOREM OF PROJECTIVE GEOMETRY

Abstract

On Characteristic Polynomials and the Norm Mapping for Dickson Near-Fields

For any (left) near-field \((F,+, \cdot)\) and a ∈ F the left multiplication \(\lambda_a : x \mapsto a \cdot x (x \in F)\) is an endomorphism of the right vector space (F, KF), where KF denotes the kernel of \((F,+, \cdot)\). In case that K is a commutative subfield of KF such that dim(F, K) is finite, it makes sense to investigate the characteristic polynomial of λa and, in particular, the norm of a over K. In this note we study these objects for the class of Dickson near-fields which are coupled to commutative fields. We show how characteristic polynomials and norms of such a near-field can be derived from those of the underlying field.

Ordered Symmetric Minkowski Planes I

Let \({\mathcal{M}}\) be a symmetric Minkowski plane. By [1, 2, 6] there is a commutative field F such that the circles of \({\mathcal{M}}\) can be identified with the elements of the projective linear group PGL(2, F). We consider only the case \(char {\mathcal{M}} := char F \neq 2\). Two circles A, B are orthogonal (\(A {\perp}B\)) iff A ≠ B and A−1B = B−1A. We introduce in \({\mathcal{M}}\) an order structure by a valuation on the set Λ of circles : “[ ] : Λ → {1, −1}” satisfying the condition : “ $$ {\rm If}\, A,B,C {\epsilon}{\Lambda}\,{\rm with} A {\perp B,C}\,{\rm then} [A] · [B] · [C] · [B · \,A^{-1} ·C] = 1.$$ ” We show that there is a one to one correspondence with these order structures of \({\mathcal{M}}\) and the halforders of the field F. In a forthcoming paper the notion of “separation” for quadruples of concyclic points will be derived from a valuation and so the connection with H.-J. Kroll’s (cf. [9]) concept of an ordered symmetric Minkowski plane established.

On the general solution of a generalization of the Goła̧b–Schinzel equation

Let X be a linear space over a commutative field K. Under some additional assumptions we determine a description of the general solution of the equation $$f(x + M(f(x))y) = f(x) \circ f(y)$$ , where $$f : X \rightarrow K, M : K \rightarrow K and \circ : K^{2} \rightarrow K$$ are unknown functions.

Vectors, Cyclic Submodules, and Projective Spaces Linked with Ternions

Given a ring of ternions $R$, i. e., a ring isomorphic to that of upper triangular $2\times 2$ matrices with entries from an arbitrary commutative field $F$, a complete classification is performed of the vectors from the free left $R$-module $R^{n+1}$, $n \geq 1$, and of the cyclic submodules generated by these vectors. The vectors fall into $5 + |F|$ and the submodules into 6 distinct orbits under the action of the general linear group $\GL_{n+1}(R)$. Particular attention is paid to {\it free} cyclic submodules generated by \emph{non}-unimodular vectors, as these are linked with the lines of $\PG(n,F)$, the $n$-dimensional projective space over $F$. In the finite case, $F$ = $\GF(q)$, explicit formulas are derived for both the total number of non-unimodular free cyclic submodules and the number of such submodules passing through a given vector. These formulas yield a combinatorial approach to the lines and points of $\PG(n,q)$, $n\geq 2$, in terms of vectors and non-unimodular free cyclic submodules of $R^{n+1}$.

On the product of vector spaces in a commutative field extension

Let K ⊂ L be a commutative field extension. Given K-subspaces A , B of L, we consider the subspace 〈 A B 〉 spanned by the product set A B = { a b | a ∈ A , b ∈ B } . If dim K A = r and dim K B = s , how small can the dimension of 〈 A B 〉 be? In this paper we give a complete answer to this question in characteristic 0, and more generally for separable extensions. The optimal lower bound on dim K 〈 A B 〉 turns out, in this case, to be provided by the numerical function κ K , L ( r , s ) = min h ( ⌈ r / h ⌉ + ⌈ s / h ⌉ − 1 ) h , where h runs over the set of K-dimensions of all finite-dimensional intermediate fields K ⊂ H ⊂ L . This bound is closely related to one appearing in additive number theory.

Noncommutative associative superproduct for general supersymplectic forms

We define a noncommutative and nonanticommutative associative product for general supersymplectic forms, allowing the explicit treatment of non(anti)commutative field theories from general nonconstant string backgrounds like a graviphoton field. We propose a generalization of deformation quantization a la Fedosov to superspace, which considers noncommutativity in the tangent bundle instead of base space, by defining the Weyl super product of elements of Weyl super algebra bundles. Super Poincare symmetry is not broken and chirality seems not to be compromised in our formulation. We show that, for a particular case, the projection of the Weyl super product to the base space gives rise the Moyal product for non(anti)commutative theories.

Correspondence between Poincaré symmetry of commutative QFT and twisted Poincaré symmetry of noncommutative QFT

The space-time symmetry of noncommutative quantum field theories with a deformed quantization is described by the twisted Poincar\'e algebra, while that of standard commutative quantum field theories is described by the Poincar\'e algebra. Based on the equivalence of the deformed theory with a commutative field theory, the correspondence between the twisted Poincar\'e symmetry of the deformed theory and the Poincar\'e symmetry of a commutative theory is established. As a by-product, we obtain the conserved charge associated with the twisted Poincar\'e transformation to make the twisted Poincar\'e symmetry evident in the deformed theory. Our result implies that the equivalence between the commutative theory and the deformed theory holds in a deeper level, i.e., it holds not only in correlation functions but also in (different types of) symmetries.

Scalar products of elementary distributions

The Schrödinger equation with point interaction in one dimension is revisited in a simple framework where the “singular” potential is defined as a symmetric operator in a natural way. The main tool is a scalar product of the elementary distributions constructed after a commutative (and well-ordered) field extension of R followed by complexification. A contact with the hyper-real numbers that arise in nonstandard analysis is possible but not essential, our extensions of R and C being obtained by a quite elementary method.

Noncommutative fields and actions of twisted Poincaré algebra

Within the context of the twisted Poincaré algebra, there exists no noncommutative analog of the Minkowski space interpreted as the homogeneous space of the Poincaré group quotiented by the Lorentz group. The usual definition of commutative classical fields as sections of associated vector bundles on the homogeneous space does not generalize to the noncommutative setting, and the twisted Poincaré algebra does not act on noncommutative fields in a canonical way. We make a tentative proposal for the definition of noncommutative classical fields of any spin over the Moyal space, which has the desired representation theoretical properties. We also suggest a way to search for noncommutative Minkowski spaces suitable for studying noncommutative field theory with deformed Poincaré symmetries.

Parametric Representation of “Covariant” Noncommutative QFT Models

We extend the parametric representation of renormalizable non commutative quantum field theories to a class of theories which we call critical, because their power counting is definitely more difficult to obtain. This class of theories is important since it includes gauge theories, which should be relevant for the quantum Hall effect.

On the stability of the quadratic equation on groups

In this paper the stability of the quadratic equation is considered on arbitrary groups. Since the quadratic equation is stable on Abelian groups, this paper examines the stability of the quadratic equation on noncommutative groups. It is shown that the quadratic equation is stable on $n$-Abelian groups when $n$ is a positive integer. The stability of the quadratic equation is also established on the noncommutative group $T(2, K)$, where $K$ is an arbitrary commutative field. It is proved that every group can be embedded into a group in which the quadratic equation is stable.

Duality and braiding in twisted quantum field theory

We re-examine various issues surrounding the definition of twisted quantum field theories on flat noncommutative spaces. We propose an interpretation based on nonlocal commutative field redefinitions which clarifies previously observed properties such as the formal equivalence of Green's functions in the noncommutative and commutative theories, causality, and the absence of UV/IR mixing. We use these fields to define the functional integral formulation of twisted quantum field theory. We exploit techniques from braided tensor algebra to argue that the twisted Fock space states of these free fields obey conventional statistics. We support our claims with a detailed analysis of the modifications induced in the presence of background magnetic fields, which induces additional twists by magnetic translation operators and alters the effective noncommutative geometry seen by the twisted quantum fields. When two such field theories are dual to one another, we demonstrate that only our braided physical states are covariant under the duality.

Smith normal form of a matrix of generalized polynomials with rational exponents

It is proved that generalized polynomials with rational exponents over a commutative field form an elementary divisor ring; an algorithm for computing the Smith normal form is derived and implemented.