Commutative Theory Research Articles

Actions of small groups on two-dimensional Artin-Schelter regular algebras

In commutative invariant theory, a classical result due to Auslander says that if R=k[x1,…,xn] and G is a finite subgroup of Autgr(R)≅GL(n,k) which contains no reflections, then there is a natural graded isomorphism R#G≅EndRG(R). In this paper, we show that a version of Auslander's Theorem holds if we replace R by an Artin-Schelter regular algebra A of global dimension 2, and G by a finite subgroup of Autgr(A) which contains no quasi-reflections. This extends work of Chan–Kirkman–Walton–Zhang. As part of the proof, we classify all such pairs (A,G), up to conjugation of G by an element of Autgr(A). In all but one case, we also write down explicit presentations for the invariant rings AG, and show that they are isomorphic to factors of AS regular algebras.

Generalizations of some classical theorems to D-normal operators on Hilbert spaces

We say that a Drazin invertible operator T on Hilbert space is of class [DN] if T^{D}T^{*} = T^{*}T^{D}. The authors in (Oper. Matrices 12(2):465–487, 2018) studied several properties of this class. We prove the Fuglede–Putnam commutativity theorem for D-normal operators. Also, we show that T has the Bishop property (beta). Finally, we generalize a very famous result on products of normal operators due to I. Kaplansky to D-normal matrices.

An Arcless Theory of Mechanical–Electronic Hybrid Commutation

The mechanical commutator of a brushed dc motor has a risk of arcing, and the electronic commutator of a brushless dc (BLdc) motor has a high cost and complicated control. In this article, a novel topology for mechanical–electronic hybrid commutation is proposed, and a hybrid commutation theory considering contact resistance is established. In the hybrid commutator, the mechanical component consisting of a commutator and three brushes is used to generate three-phase 120 $^{\circ }$ square waves. The electronic component is a freewheeling circuit consisting of a capacitor, inductors, and diodes that can suppress the arcing caused by the mechanical component. In contrast with the mechanical commutation of a brushed dc motor, the hybrid commutator fundamentally solves the problems of the interrupting arc and closing arc. Compared with the electronic commutation of a BLdc motor, the hybrid commutator does not require power electronic switches and position sensors and has a low cost and simple control. The validity of the proposed hybrid commutation theory is verified by experiments and the practicality is discussed.

Algebraic techniques for least squares problems in commutative quaternionic theory

Due to the rise of commutative quaternion in Hopfield neural networks, digital signal, and image processing, one encounters the approximate solution problems of the commutative quaternion linear equations and . This paper, by means of real representation and complex representation of commutative quaternion matrices, introduces concepts of norms of commutative quaternion matrices and derives two algebraic techniques for finding solutions of least squares problems in commutative quaternionic theory.

Toeplitz Operators on the Symmetrized Bidisc

Abstract The symmetrized bidisc has been a rich field of holomorphic function theory and operator theory. A certain well-known reproducing kernel Hilbert space of holomorphic functions on the symmetrized bidisc resembles the Hardy space of the unit disc in several aspects. This space is known as the Hardy space of the symmetrized bidisc. We introduce the study of those operators on the Hardy space of the symmetrized bidisc that are analogous to Toeplitz operators on the Hardy space of the unit disc. More explicitly, we first study multiplication operators on a bigger space (an $L^2$-space) and then study compressions of these multiplication operators to the Hardy space of the symmetrized bidisc and prove the following major results. (1) Theorem I analyzes the Hardy space of the symmetrized bidisc, not just as a Hilbert space, but as a Hilbert module over the polynomial ring and finds three isomorphic copies of it as $\mathbb D^2$-contractive Hilbert modules. (2) Theorem II provides an algebraic, Brown and Halmos-type characterization of Toeplitz operators. (3) Theorem III gives several characterizations of an analytic Toeplitz operator. (4) Theorem IV characterizes asymptotic Toeplitz operators. (5) Theorem V is a commutant lifting theorem. (6) Theorem VI yields an algebraic characterization of dual Toeplitz operators. Every section from Section 2 to Section 7 contains a theorem each, the main result of that section.

On projective intersection graph of ideals of commutative rings

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] the set of all nontrivial proper ideals of [Formula: see text]. The intersection graph of ideals of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as the set [Formula: see text], and, for any two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we study some connections between commutative ring theory and graph theory by investigating topological properties of intersection graph of ideals. In particular, it is shown that for any nonlocal Artinian ring [Formula: see text], [Formula: see text] is a projective graph if and only if [Formula: see text] where [Formula: see text] is a local principal ideal ring with maximal ideal [Formula: see text] of nilpotency three and [Formula: see text] is a field. Furthermore, it is shown that for an Artinian ring [Formula: see text] [Formula: see text] if and only if [Formula: see text] where each [Formula: see text] is a local principal ideal ring with maximal ideal [Formula: see text] such that [Formula: see text]

Two Weight Commutators on Spaces of Homogeneous Type and Applications

In this paper, we establish the two weight commutator theorem of Calderon–Zygmund operators in the sense of Coifman–Weiss on spaces of homogeneous type, by studying the weighted Hardy and BMO space for $$A_2$$ weights and by proving the sparse operator domination of commutators. The main tool here is the Haar basis, the adjacent dyadic systems on spaces of homogeneous type, and the construction of a suitable version of a sparse operator on spaces of homogeneous type. As applications, we provide a two weight commutator theorem (including the high order commutators) for the following Calderon–Zygmund operators: Cauchy integral operator on $${\mathbb {R}}$$ , Cauchy–Szego projection operator on Heisenberg groups, Szego projection operators on a family of unbounded weakly pseudoconvex domains, the Riesz transform associated with the sub-Laplacian on stratified Lie groups, as well as the Bessel Riesz transforms (in one and several dimensions).

Algebraic Theories and Commutativity in a Sheaf Topos

For any site of definition $$\mathcal {C}$$ C of a Grothendieck topos $$\mathcal {E}$$ E, we define a notion of a $$\mathcal {C}$$ C-ary Lawvere theory $$\tau : \mathscr {C} \rightarrow \mathscr {T}$$ τ:C→T whose category of models is a stack over $$\mathcal {E}$$ E. Our definitions coincide with Lawvere’s finitary theories when $$\mathcal {C}=\aleph _0$$ C=ℵ0 and $$\mathcal {E} = {{\,\mathrm{\mathbf {Set}}\,}}$$ E=Set. We construct a fibered category $${{\,\mathrm{\mathbf {Mod}}\,}}^{\mathscr {T}}$$ ModT of models as a stack over $$\mathcal {E}$$ E and prove that it is $$\mathcal {E}$$ E-complete and $$\mathcal {E}$$ E-cocomplete. We show that there is a free-forget adjunction $$F \dashv U: {{\,\mathrm{\mathbf {Mod}}\,}}^{\mathscr {T}} \leftrightarrows \mathscr {E}$$ F⊣U:ModT⇆E. If $$\tau $$ τ is a commutative theory in a certain sense, then we obtain a “locally monoidal closed” structure on the category of models, which enhances the free-forget adjunction to an adjunction of symmetric monoidal $$\mathcal {E}$$ E-categories. Our results give a general recipe for constructing a monoidal $$\mathcal {E}$$ E-cosmos in which one can do enriched $$\mathcal {E}$$ E-category theory. As an application, we describe a convenient category of linear spaces generated by the theory of Lebesgue integration.

Commutative–non-commutative dualities

We show that it is in principle possible to construct dualities between commutative and non-commutative theories in a systematic way. This construction exploits a generalization of the exact renormalization group equation (ERG). We apply this to the simple case of the Landau problem and then generalize it to the free and interacting non-canonical scalar field theory. This constructive approach offers the advantage of tracking the implementation of the Lorentz symmetry in the non-commutative dual theory. In principle, it allows for the construction of completely consistent non-commutative and non-local theories where the Lorentz symmetry and unitarity are still respected, but may be implemented in a highly non-trivial and non-local manner.

Global Seiberg–Witten Maps for U(n)-Bundles on Tori and T-duality

Seiberg-Witten maps are a well-established method to locally construct noncommutative gauge theories starting from commutative gauge theories. We revisit and classify the ambiguities and the freedom in the definition. Geometrically, Seiberg-Witten maps provide a quantization of bundles with connections. We study the case of U(n)-vector bundles on two-dimensional tori, prove the existence of globally defined Seiberg-Witten maps (induced from the plane to the torus) and show their compatibility with Morita equivalence.

Supernilpotence need not imply nilpotence

Supernilpotence is a generalization of nilpotence using a recently developed theory of higher-arity commutators for universal algebras. Many important structural properties have been shown to be associated with supernilpotence, and the exact relationship between nilpotence and supernilpotence has been the subject of investigation. We construct an algebra which is not solvable (and hence not nilpotent) but which is supernilpotent, thereby showing that in general supernilpotence does not imply nilpotence. We also extend this construction to ‘higher dimensions’ to obtain similar results for (n)-step supernilpotence.

Two applications of Nagata rings and modules

Let [Formula: see text] be a finite type hereditary torsion theory on the category of all modules over a commutative ring. The purpose of this paper is to give two applications of Nagata rings and modules in the sense of Jara [Nagata rings, Front. Math. China 10 (2015) 91–110]. First they are used to obtain Chase’s Theorem for [Formula: see text]-coherent rings. In particular, we obtain the [Formula: see text]-version of Chase’s Theorem, where [Formula: see text] is the classical star operation in ideal theory. In the second half, we apply they to characterize [Formula: see text]-flatness in the sense of Van Oystaeyen and Verschoren [Relative Invariants of Rings-The Commutative Theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 79 (Marcel Dekker, Inc., New York, 1983)].

Normal pairs of noncommutative rings

This paper extends the concept of a normal pair from commutative ring theory to the context of a pair of (associative unital) rings. This is done by using the notion of integrality introduced by Atterton. It is shown that if $$R \subseteq S$$ are rings and $$D=(d_{ij})$$ is an $$n\times n$$ matrix with entries in S, then D is integral (in the sense of Atterton) over the full ring of $$n\times n$$ matrices with entries in R if and only if each $$d_{ij}$$ is integral over R. If $$R \subseteq S$$ are rings with corresponding full rings of $$n\times n$$ matrices $$R_n$$ and $$S_n$$, then $$(R_n,S_n)$$ is a normal pair if and only if (R, S) is a normal pair. Examples are given of a pair $$(\Lambda , \Gamma )$$ of noncommutative (in fact, full matrix) rings such that $$\Lambda \subset \Gamma $$ is (resp., is not) a minimal ring extension; it can be further arranged that $$(\Lambda , \Gamma )$$ is a normal pair or that $$\Lambda \subset \Gamma $$ is an integral extension.

‐Bootstrap Approach to Non‐Commutative Gauge Theories

AbstractA consistent description of gauge theories on coordinate dependent non‐commutative (NC) space‐time is a long‐standing problem with a number of solutions, none of which is free from criticism. In this work, we discuss the approach proposed in Blumenhagen et al. [1] based on the conjecture that any consistent gauge theory can be described in terms of the ‐structure. Starting with a well‐defined commutative gauge theory, we represent it, together with the non‐commutative deformation, as a part of a bigger ‐algebra by setting some initial brackets ℓ1, ℓ2, etc. Then, solving the ‐relations we determine the missing brackets and close the ‐algebra defining the NC gauge theory which reproduces in the commutative limit the original one. We provide the recurrence relations for the construction of the pure gauge algebra , using which we find an explicit form of the NC ‐like and non‐associative octonionic‐like deformations of the Abelian gauge transformations. The construction of the ‐algebra describing the dynamics is discussed using the example of the NC Chern–Simons theory. The obtained equations of motion are non‐Lagrangian, which indicates the difference between our approach and the previous ones.

Quantum theory in quaternionic Hilbert space: How Poincaré symmetry reduces the theory to the standard complex one

As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.

Directly from H-flux to the family of three nonlocal R-flux theories

In this article we consider T-dualization of the 3D closed bosonic string in the weakly curved background — constant metric and Kalb-Ramond field with one nonzero component, Bxy = Hz, where field strength H is infinitesimal. We use standard and generalized Buscher T-dualization procedure and make T-dualization starting from coordinate z, via y and finally along x coordinate. All three theories are nonlocal, because variable ΔV, defined as line integral, appears as an argument of background fields. After the first T-dualization we obtain commutative and associative theory, while after we T-dualize along y, we get, κ-Minkowski-like, noncommutative and associative theory. At the end of this T-dualization chain we come to the theory which is both noncommutative and nonassociative. The form of the final T-dual action does not depend on the order of T-dualization while noncommutativity and nonassociativity relations could be obtained from those in the x → y → z case by replacing H → −H.

Commutativity Theorems in Groups with Power-like Maps

There are several commutativity theorems in groups and rings which involve power maps f(x) = x n . The most famous example of this kind is Jacobson's theorem which asserts that any ring satisfying the identity x n = x is commutative. Such statements belong to first order logic with equality and hence provable, in principle, by any first-order theorem-prover. However, because of the presence of an arbitrary integer parameter n in the exponent, they are outside the scope of any first-order theorem-prover. In particular, one cannot use such an automated reasoning system to prove theorems involving power maps. Here we focus just on the needed properties of power maps f(x) = x n and show how one can avoid having to reason explicitly with integer exponents. Implementing these new equational properties of power maps, we show how a theorem-prover can be a handy tool for quickly proving or confirming the truth of such theorems.

Some commutativity theorems for rings with involution involving generalized derivations

Our purpose in this paper is to investigate commutativity of a ring with involution [Formula: see text] which admits a generalized derivation satisfying certain algebraic identities. Some well-known results characterizing commutativity of prime rings have been generalized. Moreover, we provide examples to show that the assumed restrictions cannot be relaxed.

Some Results on the Classes of D-normal Operators and n-power D-normal Operators

Let $${\mathcal {B}}({\mathcal {H}})$$ be space of all bounded linear operators on a finite complex Hilbert space $${\mathcal {H}}$$ , and $$S, T \in {\mathcal {B}}({\mathcal {H}})$$ . In this paper we investigate a necessary and sufficient condition for the D-normality of ST and TS. Also, we deduce a result relating the factors in a polar decomposition of S to the D-normality of ST and TS. Moreover, we generalize Fuglede–Putnam commutativity theorem for D-normal operators. Finally, we generalize these results when the n-power D-normal operators are considered.

On linear operators for which TTD is normal

A Drazin invertible operator T ? B(H) is called skew D-quasi-normal operator if T* and TTD commute or equivalently TTD is normal. In this paper, firstly we give a list of conditions on an operator T; each of which is equivalent to T being skew D-quasi-normal. Furthermore, we obtain the matrix representation of these operators. We also develop some basic properties of such operators. Secondly we extend the Kaplansky theorem and the Fuglede-Putnam commutativity theorem for normal operators to skew D-quasi-normal matrices.