Abelian Case Research Articles

Geometry of cascade feedback linearizable control systems

Cascade feedback linearization provides geometric insights on explicit integrability of nonlinear control systems with symmetry. A central piece of the theory requires that the partial contact curve reduction of the contact sub-connection be static feedback linearizable. This work establishes new necessary conditions on the equations of Lie type - in the abelian case - that arise in a contact sub-connection with the desired static feedback linearizability property via families of codimension one partial contact curves. Furthermore, an explicit class of contact sub-connections admitting static feedback linearizable contact curve reductions is presented, hinting at a possible classification of all such contact sub-connections. Key tools in proving, and stating, the main results of this paper are truncated versions of the total derivative and Euler operators. Additionally, the Battilotti-Califano system with three inputs is used as a clarifying example of both cascade feedback linearization and the new necessary conditions.

A massive gauge theory à la Utiyama

Utiyama’s method is a deductive approach of building gauge theories for semi-simple groups of local transformations, including the Abelian U(1) case, the non-Abelian SU(N) group, and the gravitational interaction. Gauge theories à la Utiyama typically predict a massless gauge potential. This work brings a mass generation mechanism and Utiyama’s method together thus giving mass to the interaction boson without breaking the gauge symmetry. Herein we devote our attention to the Abelian case. Two gauge potentials are introduced: a vetor field A μ and a scalar field B. The associated gauge-invariant field strengths F μ ν and G μ are built from Utiyama’s technique. Gauge invariance requirement upon the total Lagrangian (including matter fields and gauge fields) yields the conserved currents. Finally, we study the simplest type of Lagrangian involving the field strengths and obtain the related field equation. By imposing appropriate constraints on this particular example, Stueckelberg model is recovered.

Counting operators in N = 1 supersymmetric gauge theories

Following a recent publication, in this paper we count the number of independent operators at arbitrary mass dimension in N = 1 supersymmetric gauge theories and derive their field and derivative content. This work uses Hilbert series machinery and extends a technique from our previous work on handling integration by parts redundancies to vector superfields. The method proposed here can be applied to both abelian and non-abelian gauge theories and for any set of (chiral/antichiral) matter fields. We work through detailed steps for the abelian case with single flavor chiral superfield at mass dimension eight, and provide other examples in the appendices.

On integral mixed Cayley graphs over non-abelian finite groups admitting an abelian subgroup of index 2

Recently, several works by a number of authors have provided characterizations of integral undirected Cayley graphs over generalized dihedral groups and generalized dicyclic groups. We generalize and unify these results in two different ways. Firstly, we work over arbitrary non-abelian finite groups admitting an abelian subgroup of index 2. Secondly, our main result actually characterizes integral mixed Cayley graphs over such finite groups, in the spirit of a very recent result of Kadyan–Bhattarcharjya in the abelian case.

Class fields, Dirichlet characters, and extended genus fields of global function fields

AbstractWe study the extended genus field of an abelian extension of a rational function field. We follow the definition of Anglès and Jaulent, which uses the class field theory. First, we show that the natural definition of extended genus field of a cyclotomic function field obtained by means of Dirichlet characters is the same as the one given by Anglès and Jaulent. Next, we study the extended genus field of a finite abelian extension of a rational function field along the lines of the study of genus fields of abelian extensions of rational function fields. In the absolute abelian case, we compare this approach with the one given by Anglès and Jaulent.

Towards equivariant Yang-Mills theory

We study four dimensional gauge theories in the context of an equivariant extension of the Batalin-Vilkovisky (BV) formalism. We discuss the embedding of BV Yang-Mills (YM) theory into a larger BV theory and their relation. Partial integration in the equivariant BV setting (BV push-forward map) is performed explicitly for the abelian case. As result, we obtain a non-local homological generalization of the Cartan calculus and a non-local extension of the abelian YM BV action which satisfies the equivariant master equation.

Quantum scale invariance in gauge theories and applications to muon production

We discuss quantum scale invariance in (scale invariant) gauge theories with both ultraviolet (UV) and infrared (IR) divergences. Firstly, their BRST invariance is checked in two apparently unrelated approaches using a scale invariant regularisation (SIR). These approaches are then shown to be equivalent. Secondly, for the Abelian case we discuss both UV and IR quantum corrections present in such theories. We present the Feynman rules in a form suitable for offshell Green functions calculations, together with their one-loop renormalisation. This information is then used for the muon production cross section at one-loop in a quantum scale invariant theory. Such a theory contains not only new UV poles but also IR poles. While the UV poles bring new quantum corrections (in the form of counterterms), finite or divergent, that we compute, it is shown that the IR poles do not bring new physics. The IR quantum corrections, both finite and divergent, cancel out similarly to the way the IR poles themselves cancel in the traditional approach to IR divergences (in the cross section, after summing over virtual and real corrections). Hence, the evanescent interactions induced by the scale-invariant analytical continuation of the SIR scheme do not affect IR physics, as illustrated at one-loop for the muon production ($e^+ e^- \to \mu^+\mu^-$) cross section.

On the functional graph of the power map over finite groups

In this paper we study the description of the functional graphs associated with the power maps over finite groups. We present a structural result which describes the isomorphism class of these graphs for abelian groups and also for flower groups, which is a special class of non abelian groups introduced in this paper. Unlike the abelian case where all the trees associated with periodic points are isomorphic, in the case of flower groups we prove that several different classes of trees can occur. The class of central trees (i.e. associated with periodic points that are in the center of the group) are in general non-elementary and a recursive description is given in this work. Flower groups include many non abelian groups such as dihedral and generalized quaternion groups, and the projective general linear group of order two over a finite field. In particular, we provide improvements on past works regarding the description of the dynamics of the power map over these groups.

Step towards a consistent treatment of chiral theories at higher loop order: The abelian case

As recently pointed out, regularization schemes defined in four-dimensions may also face inconsistencies in the presence of chiral fermions. In this work, we extend this analysis to two-loop order. Adopting the implicit regularization as working arena, we discuss in detail how a consistent version of the method can be envisaged, and present many of the subtleties that appear at two-loop order using an abelian chiral model as working example.

Affine nil-Hecke algebras and quantum cohomology

Let G be a compact, connected Lie group and T⊂G a maximal torus. Let (M,ω) be a monotone closed symplectic manifold equipped with a Hamiltonian action of G. We construct a module action of the affine nil-Hecke algebra Hˆ⁎S1×T(LG/T) on the S1×T-equivariant quantum cohomology of M, QHS1×T⁎(M). Our construction generalizes the theory of shift operators for Hamiltonian torus actions [46,40]. We show that, as in the abelian case, this action behaves well with respect to the quantum connection. As an application of our construction, we show that the G-equivariant quantum cohomology QHG⁎(M) defines a canonical holomorphic Lagrangian subvariety LG(M)↪BFM(GC∨) in the BFM-space of the Langlands dual group, confirming an expectation of Teleman from [51].

Unlikely intersections of curves with algebraic subgroups in semiabelian varieties

Let G be a semiabelian variety and C a curve in G that is not contained in a proper algebraic subgroup of G. In this situation, conjectures of Pink and Zilber imply that there are at most finitely many points contained in the so-called unlikely intersections of C with subgroups of codimension at least 2. In this note, we establish this assertion for general semiabelian varieties over overline{mathbb {Q}}. This extends results of Maurin and Bombieri, Habegger, Masser, and Zannier in the toric case as well as Habegger and Pila in the abelian case.

Non-Abelian chiral kinetic equations in the Cartan-Weyl basis

<sec>Non-Abelian gauge field is the fundamental element of the standard model. Non-Abelian chiral kinetic theory can be used to describe how the chiral fermions in standard model transport in a non-equilibrium system. </sec><sec>In our previous work, we decomposed the non-Abelian chiral kinetic equations into color singlet and multiplet in the <inline-formula><tex-math id="M1">\begin{document}$SU(N)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M1.png"/></alternatives></inline-formula> color space. In this formalism, the chiral kinetic equations preserve the gauge symmetry in a very apparent way. However, sometimes we need to describe the microscopic process of the specific color degree, e.g. the color connection in the hadronization stage. In order to describe such a process, it will be more convenient to decompose the non-Abelian chiral kinetic equations in the Cartan-Weyl basis. </sec><sec>In this work, we choose the matrix elements of the Wigner function in fundamental representation of color space as the direct variables and decompose the gauge field or strength tensor field in the Cartan-Weyl basis. By using the covariant gradient expansion, we decompose the non-Abelian chiral kinetic equations into the coupled kinetic equations for diagonal distribution function and non-diagonal distribution function up to the first order. When only diagonal elements exist in the gauge field with non-diagonal elements and diagonal elements decoupled, the non-Ableian chiral kinetic equation will be reduced to the form in the Abelian case. When the non-diagonal elements of the gauge field are present, the kinetic equations are totally tangled between diagonal distribution function and non-diagonal distribution function. Especially, the <inline-formula><tex-math id="M2">\begin{document}$0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M2.png"/></alternatives></inline-formula>th-order non-diagonal distribution function could induce the <inline-formula><tex-math id="M3">\begin{document}$1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M3.png"/></alternatives></inline-formula>st-order diagonal Wigner function, and the <inline-formula><tex-math id="M4">\begin{document}$0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M4.png"/></alternatives></inline-formula>th-order diagonal distribution function could also induce the <inline-formula><tex-math id="M5">\begin{document}$1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M5.png"/></alternatives></inline-formula>st-order non-diagonal Wigner function. </sec>

The topological entropy of powers on Lie groups

AbstractIn this paper, we address the problem of computing the topological entropy of a map $\psi : G \to G$ , where G is a Lie group, given by some power $\psi (g) = g^k$ , with k a positive integer. When G is abelian, $\psi $ is an endomorphism and its topological entropy is given by $h(\psi ) = \dim (T(G)) \log (k)$ , where $T(G)$ is the maximal torus of G, as shown by Patrão [The topological entropy of endomorphisms of Lie groups. Israel J. Math.234 (2019), 55–80]. However, when G is not abelian, $\psi $ is no longer an endomorphism and these previous results cannot be used. Still, $\psi $ has some interesting symmetries, for example, it commutes with the conjugations of G. In this paper, the structure theory of Lie groups is used to show that $h(\psi ) = \dim (T)\log (k)$ , where T is a maximal torus of G, generalizing the formula in the abelian case. In particular, the topological entropy of powers on compact Lie groups with discrete center is always positive, in contrast to what happens to endomorphisms of such groups, which always have null entropy.

Iterated Minkowski sums, horoballs and north-south dynamics

Given a finite generating set A for a group \Gamma , we study the map W \mapsto WA as a topological dynamical system – a continuous self-map of the compact metrizable space of subsets of \Gamma . If the set A generates \Gamma as a semigroup and contains the identity, there are precisely two fixed points, one of which is attracting. This supports the initial impression that the dynamics of this map is rather trivial. Indeed, at least when \Gamma= \mathbb{Z}^d and A \subseteq \mathbb{Z}^d is a finite positively generating set containing the identity, the natural invertible extension of the map W \mapsto W+A is always topologically conjugate to the unique “north-south” dynamics on the Cantor set. In contrast to this, we show that various natural “geometric” properties of the finitely generated group (\Gamma,A) can be recovered from the dynamics of this map, in particular, the growth type and amenability of \Gamma . When \Gamma = \mathbb{Z}^d , we show that the volume of the convex hull of the generating set A is also an invariant of topological conjugacy. Our study introduces, utilizes and develops a certain convexity structure on subsets of the group \Gamma , related to a new concept which we call the sheltered hull of a set. We also relate this study to the structure of horoballs in finitely generated groups, focusing on the abelian case.

Weyl-Einstein structures on conformal solvmanifolds

A conformal Lie group is a conformal manifold (M, c) such that M has a Lie group structure and c is the conformal structure defined by a left-invariant metric g on M. We study Weyl-Einstein structures on conformal solvable Lie groups and on their compact quotients. In the compact case, we show that every conformal solvmanifold carrying a Weyl-Einstein structure is Einstein. We also show that there are no left-invariant Weyl-Einstein structures on non-abelian nilpotent conformal Lie groups, and classify them on conformal solvable Lie groups in the almost abelian case. Furthermore, we determine the precise list (up to automorphisms) of left-invariant metrics on simply connected solvable Lie groups of dimension 3 carrying left-invariant Weyl-Einstein structures.

Holomorphic Reflexivity for Locally Finite and Profinite Groups: The Abelian and General Cases

Holomorphic Reflexivity for Locally Finite and Profinite Groups: The Abelian and General Cases

Wave Packet Dynamics in Synthetic Non-Abelian Gauge Fields.

It is generally admitted that in quantum mechanics, the electromagnetic potentials have physical interpretations otherwise absent in classical physics as illustrated by the Aharonov-Bohm effect. In 1984, Berry interpreted this effect as a geometrical phase factor. The same year, Wilczek and Zee generalized the concept of Berry phases to degenerate levels and showed that a non-Abelian gauge field arises in these systems. In sharp contrast with the Abelian case, spatially uniform non-Abelian gauge fields can induce particle noninertial motion. We explore this intriguing phenomenon with a degenerated Fermionic atomic gas subject to a two-dimensional synthetic SU(2) non-Abelian gauge field. We reveal the spin Hall nature of the noninertial dynamic as well as its anisotropy in amplitude and frequency due to the spin texture of the system. We finally draw the similarities and differences of the observed wave packet dynamic and the celebrated Zitterbewegung effect of the relativistic Dirac equation.

Chromatic symmetric functions of Dyck paths and [formula omitted]-rook theory

The chromatic symmetric function (CSF) of Dyck paths of Stanley and its Shareshian–Wachs q-analogue have important connections to Hessenberg varieties, diagonal harmonics and LLT polynomials. In the, so called, abelian case they are also curiously related to placements of non-attacking rooks by results of Stanley and Stembridge (1993) and Guay-Paquet (2013). For the q-analogue, these results have been generalized by Abreu and Nigro (2021) and Guay-Paquet (private communication), using q-hit numbers. Among our main results is a new proof of Guay-Paquet’s elegant identity expressing the q-CSFs in a CSF basis with q-hit coefficients. We further show its equivalence to the Abreu–Nigro identity expanding the q-CSF in the elementary symmetric functions. In the course of our work we establish that the q-hit numbers in these expansions differ from the originally assumed Garsia–Remmel q-hit numbers by certain powers of q. We prove new identities for these q-hit numbers, and establish connections between the three different variants.

A Study on Sylow Theorems for Finding out Possible Subgroups of a Group in Different Types of Order

This paper aims at treating a study on Sylow theorem of different algebraic structures as groups, order of a group, subgroups, along with the associated notions of automorphisms group of the dihedral groups, split extensions of groups and vector spaces arises from the varying properties of real and complex numbers. We must have used the Sylow theorems of this work when it's generalized. Here we discuss possible subgroups of a group in different types of order which will give us a practical knowledge to see the applications of the Sylow theorems. In algebraic structures, we deal with operations of addition and multiplication and in order structures, those of greater than, less than and so on. It is through the study of Sylow theorems that we realize the importance of some definitions as like as the exact sequences and split extensions of groups, Sylow p-subgroup and semi-direct product. Thus it has been found necessary and convenient to study these structures in detail. In situations, where it was found that a given situation satisfies the basic axioms of structure and having already known the properties of that structure. Finally, we find out possible subgroups of a group in different types of order for abelian and non-abelian cases.

A Formalization of Finite Group Theory

Previous formulations of group theory in ACL2 and Nqthm, based on either "encapsulate" or "defn-sk", have been limited by their failure to provide a path to proof by induction on the order of a group, which is required for most interesting results in this domain beyond Lagrange's Theorem (asserting the divisibility of the order of a group by that of a subgroup). We describe an alternative approach to finite group theory that remedies this deficiency, based on an explicit representation of a group as an operation table. We define a "defgroup" macro for generating parametrized families of groups, which we apply to the additive and multiplicative groups of integers modulo n, the symmetric groups, arbitrary quotient groups, and cyclic subgroups. In addition to a proof of Lagrange's Theorem, we present an inductive proof of the abelian case of a theorem of Cauchy: If the order of a group G is divisible by a prime p, then G has an element of order p.