Nilpotency Property Research Articles

Superfield Approach to Nilpotency and Absolute Anticommutativity of Conserved Charges: 2D Non-Abelian 1-Form Gauge Theory

We exploit the theoretical strength of augmented version of superfield approach (AVSA) to Becchi-Rouet-Stora-Tyutin (BRST) formalism to express the nilpotency and absolute anticommutativity properties of the (anti-)BRST and (anti-)co-BRST conserved charges for the two (1+1)-dimensional (2D) non-Abelian 1-form gauge theory (without any interaction with matter fields) in the language of superspace variables, their derivatives, and suitable superfields. In the proof of absolute anticommutativity property, we invoke the strength of Curci-Ferrari (CF) condition for the (anti-)BRST charges. No such outside condition/restriction is required in the proof of absolute anticommutativity of the (anti-)co-BRST conserved charges. The latter observation (as well as other observations) connected with (anti-)co-BRST symmetries and corresponding conserved charges are novel results of our present investigation. We also discuss the (anti-)BRST and (anti-)co-BRST symmetry invariance of the appropriate Lagrangian densities within the framework of AVSA. In addition, we dwell a bit on the derivation of the above fermionic (nilpotent) symmetries by applying the AVSA to BRST formalism, where only the (anti)chiral superfields are used.

On the Rost nilpotence theorem for threefolds

We show that Rost nilpotence holds for a geometrically integral threefold X over a field k of characteristic 0 if and only if αk(X)*∘N( CH 0(Xk(X)))=0 for some integer N>0 for all correspondences α of degree 0 which vanish over some field extension of k. As a corollary we get the Rost nilpotence property for three-dimensional smooth projective geometrically integral schemes over a field of characteristic zero, which are birationally isomorphic to a toric model.

Countable recognizability and nilpotency properties of groups

A class Open image in new window of groups is said to be countably recognizable if a group belongs to Open image in new window whenever all its countable subgroups lie in Open image in new window. It is proved here that most of the relevant group classes of generalized nilpotent groups are countably recognizable. Moreover, some further countably recognizable group classes are exhibited.

Nilpotent elements and nil-Armendariz property of skew generalized power series rings

Let [Formula: see text] be a ring, [Formula: see text] a strictly ordered monoid, and [Formula: see text] a monoid homomorphism. The skew generalized power series ring [Formula: see text] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev–Neumann Laurent series rings. In this paper, we introduce and study the [Formula: see text]-nil-Armendariz condition on [Formula: see text], a generalization of the standard nil-Armendariz condition from polynomials to skew generalized power series. We resolve the structure of [Formula: see text]-nil-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be [Formula: see text]-nil-Armendariz. The [Formula: see text]-nil-Armendariz condition is connected to the question of whether or not a skew generalized power series ring [Formula: see text] over a nil ring [Formula: see text] is nil, which is related to a question of Amitsur [Algebras over infinite fields, Proc. Amer. Math. Soc. 7 (1956) 35–48]. As particular cases of our general results we obtain several new theorems on the nil-Armendariz condition. Our results extend and unify many existing results.

Novel symmetries in an interacting 𝒩 = 2 supersymmetric quantum mechanical model

In this paper, we demonstrate the existence of a set of novel discrete symmetry transformations in the case of an interacting [Formula: see text] supersymmetric quantum mechanical model of a system of an electron moving on a sphere in the background of a magnetic monopole and establish its interpretation in the language of differential geometry. These discrete symmetries are, over and above, the usual three continuous symmetries of the theory which together provide the physical realizations of the de Rham cohomological operators of differential geometry. We derive the nilpotent [Formula: see text] SUSY transformations by exploiting our idea of supervariable approach and provide geometrical meaning to these transformations in the language of Grassmannian translational generators on a [Formula: see text]-dimensional supermanifold on which our [Formula: see text] SUSY quantum mechanical model is generalized. We express the conserved supercharges and the invariance of the Lagrangian in terms of the supervariables (obtained after the imposition of the SUSY invariant restrictions) and provide the geometrical meaning to (i) the nilpotency property of the [Formula: see text] supercharges, and (ii) the SUSY invariance of the Lagrangian of our [Formula: see text] SUSY theory.

Nilradicals of the unique product monoid rings

Armendariz rings are generalization of reduced rings, and therefore, the set of nilpotent elements plays an important role in this class of rings. There are many examples of rings with nonzero nilpotent elements which are Armendariz. Observing structure of the set of all nilpotent elements in the class of Armendariz rings, Antoine introduced the notion of nil-Armendariz rings as a generalization, which are connected to the famous question of Amitsur of whether or not a polynomial ring over a nil coefficient ring is nil. Given an associative ring [Formula: see text] and a monoid [Formula: see text], we introduce and study a class of Armendariz-like rings defined by using the properties of upper and lower nilradicals of the monoid ring [Formula: see text]. The logical relationship between these and other significant classes of Armendariz-like rings are explicated with several examples. These new classes of rings provide the appropriate setting for obtaining results on radicals of the monoid rings of unique product monoids and also can be used to construct new classes of nil-Armendariz rings. We also classify, which of the standard nilpotence properties on polynomial rings pass to monoid rings. As a consequence, we extend and unify several known results.

Solvability and nilpotency for algebraic supergroups

We study solvability, nilpotency and splitting property for algebraic supergroups over an arbitrary field K of characteristic charK≠2. Our first main theorem tells us that an algebraic supergroup G is solvable if the associated algebraic group Gev is trigonalizable. To prove it we determine the algebraic supergroups G such that dim⁡Lie(G)1=1; their representations are studied when Gev is diagonalizable. The second main theorem characterizes nilpotent connected algebraic supergroups. A super-analogue of the Chevalley Decomposition Theorem is proved, though it must be in a weak form. An appendix is given to characterize smooth Noetherian superalgebras as well as smooth Hopf superalgebras.

Comments on the nilpotent constraint of the goldstino superfield

Superfield constraints were often used in the past, in particular, to describe the Akulov–Volkov action of the goldstino by a superfield formulation with [Formula: see text] endowed with the nilpotent constraint [Formula: see text] for the goldstino superfield ([Formula: see text]). Inspired by this, such constraint is often used to define the goldstino superfield even in the presence of additional superfields, for example, in models of “nilpotent inflation”. In this review, we show that the nilpotent property is not valid, in general, under the assumption of a microscopic (ultraviolet (UV)) description of the theory with linear supermultiplets. Sometimes only weaker versions of the nilpotent relation are true such as [Formula: see text] or [Formula: see text] ([Formula: see text]) in the infrared (far below the UV scale) under the further requirement of decoupling all additional scalars (coupling to sgoldstino), something not always possible (e.g. if light scalars exist). In such cases, the weaker nilpotent property is not specific to the goldstino superfield anymore. We review the restrictions for the Kahler curvature tensor and superpotential [Formula: see text] under which [Formula: see text] remains true in infrared, assuming linear supermultiplets in the microscopic description. One can reverse the arguments to demand that the nilpotent condition, initially an infrared property, be extended even in the presence of additional superfields, but this may question the nature of supersymmetry breaking or the existence of a perturbative UV description with linear supermultiplets.

Supervariable Approach to the Nilpotent Symmetries for a Toy Model of the Hodge Theory

We exploit the standard techniques of the supervariable approach to derive the nilpotent Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry transformations for a toy model of the Hodge theory (i.e., a rigid rotor) and provide the geometrical meaning and interpretation to them. Furthermore, wealsoderive the nilpotent (anti-)co-BRST symmetry transformations for this theory within the framework of the above supervariable approach. We capture the (anti-)BRST and (anti-)co-BRST invariance of the Lagrangian of our present theory within the framework of augmented supervariable formalism. We also express the (anti-)BRST and (anti-)co-BRST charges in terms of the supervariables (obtained after the application of the (dual-)horizontality conditions and (anti-)BRST and (anti-)co-BRST invariant restrictions) to provide the geometrical interpretations for their nilpotency and anticommutativity properties. The application of the dual-horizontality condition and ensuing proper (i.e., nilpotent and absolutely anticommuting) fermionic (anti-)co-BRST symmetries are completelynovelresults in our present investigation.

Non-abelian whole gauge symmetry

The wholeness principle is analysed for non-abelian gauge symmetry. This principle states that nature acts through grouping. It says that physical laws should be derived from elds associations. At this work, we consider on the possibility of introducting a non-abelian elds set fAaIg under a common gauge parameter.A Yang-Mills extension is studied. Taking the SU(N) symmetry group with different potential elds rotating under a same group, new elds strengths are developed. They express covariant entities which are granular, collective, correlated, and not necessarily Lie algebra valued. They yield new scalars and a Lagrangian beyond Yang-Mills is obtained. Classical equations are derived and (2N + 7) equations are developed. A further step is on how such non-abelian whole symmetry is implemented at SU(N) gauge group. For this, it is studied on the algebra closure and Jacobi identities, Bianchi identities, Noether theorem, gauge xing, BRST symmetry, conservation laws, covariance, charges algebra. As result, one notices that it is installed at SU(N) symmetry independentlyon the number of involved elds. Given this consistency, Yang-Mills should not more be considered as the unique Lagrangian performed from SU(N).Introducting the BRST symmetry an invariant Leff is stablished. The BRST charge associated to the N-potential elds system is calculated and its nilpotency property obtained.Others conservations laws involving ghost scale, global charges are evalued showing that this whole symmetry extension preserve the original Yang-Mills algebra. Also the ghost number is conserved. These results imply that Yang-Mills should be understood as a pattern and not as a specic Lagrangian.Concluding, an extended Lagrangian can be constructed. It is possible to implement a non-abelian whole gauge symmetry based on a elds set fAaIg. Its physical feature is a systemic interpretation for the physical processes. Understand complexity from whole gauge symmetry.

Symbolic and Iterative Computation of Quasi-Filiform Nilpotent Lie Algebras of Dimension Nine

This paper addresses the problem of computing the family of two-filiform Lie algebra laws of dimension nine using three Lie algebra properties converted into matrix form properties: Jacobi identity, nilpotence and quasi-filiform property. The interest in this family is broad, both within the academic community and the industrial engineering community, since nilpotent Lie algebras are applied in traditional mechanical dynamic problems and current scientific disciplines. The conditions of being quasi-filiform and nilpotent are applied carefully and in several stages, and appropriate changes of the basis are achieved in an iterative and interactive process of simplification. This has been implemented by means of the development of more than thirty Maple modules. The process has led from the first family formulation, with 64 parameters and 215 constraints, to a family of 16 parameters and 17 constraints. This structure theorem permits the exhaustive classification of the quasi-filiform nilpotent Lie algebras of dimension nine with current computational methodologies.

Formula omitted] SUSY symmetries for a moving charged particle under influence of a magnetic field: Supervariable approach

We exploit the supersymmetric invariant restrictions (SUSYIRs) on the supervariables to derive the nilpotent N=2 SUSY transformations for the supersymmetric quantum mechanical model of the motion of a charged particle in the X–Y plane (where the magnetic field (Bz) is applied along the Z-direction). The supervariables are defined on a (1, 2)-dimensional supermanifold parametrized by a bosonic “time” variable t and a pair of Grassmannian variables θ and θ̄ (with θ2=θ̄2=0,θθ̄+θ̄θ=0). We take the (anti-)chiral supervariables for our purpose so that the nilpotency property of the N=2 SUSY symmetry transformations could be captured within the framework of supervariable approach. We express the Lagrangian as well as supercharges in terms of the supervariables (that are obtained after the application of the appropriate SUSYIRs) and provide geometrical basis, within the framework of our supervariable approach, for (i) the nilpotency property of the above SUSY transformations (and the corresponding supercharges), and (ii) the SUSY invariance of the Lagrangian.

A free supersymmetric system: Novel symmetries

We discuss a set of novel discrete symmetries of a free supersymmetric (SUSY) quantum-mechanical system which is the limiting case of a widely studied interacting SUSY model of a charged particle constrained to move on a sphere in the background of a Dirac magnetic monopole. The usual continuous symmetries of this model provide the physical realization of the de Rham cohomological operators of differential geometry. The interplay between the novel discrete symmetries and usual continuous symmetries leads to the physical realization of the relationship between the (co-)exterior derivatives of differential geometry. We have also exploited the supervariable approach to derive the nilpotent SUSY symmetries of the theory and provided the geometrical origin and interpretation for the nilpotency property. Ultimately, our present study (based on innate symmetries) proves that our free SUSY example is a tractable model for the Hodge theory.

On the Hirsch–Plotkin radical of stability groups

We study the stability group of subspace series of infinite dimensional vector spaces. In [1], the authors proved that when the vector space has countable dimension then the Hirsch–Plotkin radical of the stability group coincides with the set of all space automorphisms that fix a finite subseries and they conjectured that this would hold in all dimensions. We give a counter example of dimension 2ℵ0. We however show that in general the result remains true if the Hirsch–Plotkin radical is replaced by the Fitting group, the product of all the normal nilpotent subgroups of the stability group. We also show that the Hirsch–Plotkin radical has a certain strong local nilpotence property.

General [formula omitted] supersymmetric quantum mechanical model: Supervariable approach to its off-shell nilpotent symmetries

Using the supersymmetric (SUSY) invariant restrictions on the (anti-)chiral supervariables, we derive the off-shell nilpotent symmetries of the general one (0+1)-dimensional N=2 SUSY quantum mechanical (QM) model which is considered on a (1, 2)-dimensional supermanifold (parametrized by a bosonic variable t and a pair of Grassmannian variables θ and θ̄ with θ2=θ̄2=0,θθ̄+θ̄θ=0). We provide the geometrical meanings to the two SUSY transformations of our present theory which are valid for any arbitrary type of superpotential. We express the conserved charges and Lagrangian of the theory in terms of the supervariables (that are obtained after the application of SUSY invariant restrictions) and provide the geometrical interpretation for the nilpotency property and SUSY invariance of the Lagrangian for the general N=2 SUSY quantum theory. We also comment on the mathematical interpretation of the above symmetry transformations.

Nilpotent Property of Skew Generalized Power Series Rings

Let R be an NI ring with nil(R) nilpotent, (S, ≤) a strictly totally ordered monoid satisfying the condition that 0 ≤ s for all s ∈ S ,a ndω : S → End(R) be a compatible monoid homomorphism. Then the ring ((R S,≤ ,ω )) of the skew generalized power series with coefficients in R and exponents in S is a nilpotent p.p.-ring if and only if R is a nilpotent p.p.-ring. Furthermore, if R satisfies the descending chain condition on weak annihilators, then ((R S,≤ ,ω )) is a weak APP-ring if and only if R is a weak APP-ring. Consequently, some properties of the base ring R can be profitably generalized to the skew generalized power series ring ((R S,≤ ,ω )).

On Monoid Rings Over Nil Armendariz Ring

Given a ring R and a monoid M, we study the concept of so called nil-Armendariz ring relative to a monoid, which is a common generalization of nil-Armendariz rings and Armendariz rings relative to a monoid. It is done by considering the nil-Armendariz condition on a monoid ring R[M] instead of the polynomial ring R[x]. We prove that several properties transfer between R and the monoid ring R[M], in case R is nil M-Armendariz ring. We resolve the structure of nil M-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be nil M-Armendariz, unifying and generalizing a number of known Armendariz-like conditions in the special cases. In particular, we prove that every NI-ring is nil M-Armendariz, for any unique product monoid M. We also classify which of the standard nilpotence properties on polynomial rings pass to monoid rings. We provide various examples and classify how the nil M-Armendariz rings behaves under various ring extensions.

ON NIL GENERALIZED POWER SERIESWISE ARMENDARIZ RINGS

We in this note introduce a concept, so called nil generalized power serieswise Armendariz ring, that is a generalization of both S-Armendariz rings and nil power serieswise Armendariz rings. We first observe the basic properties of nil generalized power serieswise Armendariz rings, constructing typical examples. We next study the relationship between the nilpotent property of R and that of the generalized power series ring [[<TEX>$R^{S,{\leq}}$</TEX>]] whenever R is nil generalized power serieswise Armendariz.

A DESCRIPTION OF A CLASS OF FINITE SEMIGROUPS THAT ARE NEAR TO BEING MAL'CEV NILPOTENT

In this paper we continue the investigation on the algebraic structure of a finite semigroup S that is determined by its associated upper non-nilpotent graph [Formula: see text]. The vertices of this graph are the elements of S and two vertices are adjacent if they generate a semigroup that is not nilpotent (in the sense of Mal'cev). We introduce a class of semigroups in which the Mal'cev nilpotent property lifts through ideal chains. We call this the class of pseudo-nilpotent semigroups. The definition is such that the global information that a semigroup is not nilpotent induces local information, i.e. some two-generated subsemigroups are not nilpotent. It turns out that a finite monoid (in particular, a finite group) is pseudo-nilpotent if and only if it is nilpotent. Our main result is a description of pseudo-nilpotent finite semigroups S in terms of their associated graph [Formula: see text]. In particular, S has a largest nilpotent ideal, say K, and S/K is a 0-disjoint union of its connected components (adjoined with a zero) with at least two elements.

$$K$$ -theory of toric varieties revisited

After surveying higher \(K\)-theory of toric varieties, we present Totaro’s old (c. 1997) unpublished result on expressing the corresponding homotopy theory via singular cohomology. It is a higher analog of the rational Chern character isomorphism for general toric schemes. In the special case of a projective simplicial toric scheme over a regular ring one obtains a rational isomorphism between the homotopy \(K\)-theory and the direct sum of \(m\) copies of the \(K\)-theory of the ground ring, \(m\) being the number of maximal cones in the underlying fan. Apart from its independent interest, in retrospect, Totaro’s observations motivated some (old) and complement several other (very recent) results. We conclude with a conjecture on the nil-groups of affine monoid rings, extending the nilpotence property. The conjecture holds true for \(K_0\).