Lie Group Research Papers - Academia.edu (original) (raw)

2025, Archiv der Mathematik

We construct a new infinite family of models of exotic 7-spheres. These models are direct generalizations of the Gromoll-Meyer sphere. From their symmetries, geodesics and submanifolds half of them are closer to the standard 7-sphere than... more

2025, arXiv (Cornell University)

We prove the following results: let x, y be (n, n) complex matrices such that x, y, xy have no eigenvalue in ] -∞, 0] and log(xy) = log(x) + log(y). If n = 2, or if n ≥ 3 and x, y are simultaneously triangularizable, then x, y commute. In... more

2025, Potential Analysis

We present an explicit form of the subelliptic heat kernel of the intrinsic sublaplacian \Delta _{\textrm{sub}}^5Δsub5inducedbyarank5trivializablesubriemannianstructureontheEuclideansevendimensionalsphereΔ sub 5 induced by a rank 5 trivializable subriemannian structure on the Euclidean seven dimensional sphereΔsub5inducedbyarank5trivializablesubriemannianstructureontheEuclideansevendimensionalsphere\mathbb {S}^7S7.ThiscompletestheheatkernelanalysisoftrivializablesubriemannianstructuresonS 7 . This completes the heat kernel analysis of trivializable subriemannian structures onS7.Thiscompletestheheatkernelanalysisoftrivializablesubriemannianstructureson\mathbb {S}^7S7inducedbyaCliffordmoduleactiononS 7 induced by a Clifford module action onS7inducedbyaCliffordmoduleactionon\mathbb {R}^8R8.AsanapplicationwederivethespectrumofR 8 . As an application we derive the spectrum ofR8.Asanapplicationwederivethespectrumof\Delta _{\textrm{sub}}^5$$ Δ sub 5 and the Green function of the conformal sublaplacian in an explicit form.

2025, arXiv (Cornell University)

We determine the spectrum of the sub-Laplacian on pseudo H-type nilmanifolds and present pairs of isospectral but non-diffeomorphic nilmanifolds with respect to the sub-Laplacian. We observe that these pairs are also isospectral with respect to the Laplacian. More generally, our method allows us to construct an arbitrary number of isospectral but mutually non-diffeomorphic nilmanifolds. Finally, we present two nilmanifolds of different dimensions such that the short time heat trace expansions of the corresponding sub-Laplace operators coincide up to a term which vanishes to infinite order as time tends to zero.

2025, arXiv (Cornell University)

On the seven dimensional Euclidean sphere S 7 we compare two subriemannian structures with regards to various geometric and analytical properties. The first structure is called trivializable and the underlying distribution H T is induced by a Clifford module structure of R 8 . More precisely, H T is rank 4, bracket generating of step two and generated by globally defined vector fields. The distribution H Q of the second structure is of rank 4 and step two as well and obtained as the horizontal distribution in the quaternionic Hopf fibration S 3 ֒→ S 7 → S 4 . Answering a question in we first show that H Q does not admit a global nowhere vanishing smooth section. In both cases we determine the Popp measures, the intrinsic sublaplacians ∆ T sub and ∆ Q sub and the nilpotent approximations. We conclude that both subriemannian structures are not locally isometric and we discuss properties of the isometry group. By determining the first heat invariant of the sublaplacians it is shown that both structures are also not isospectral in the subriemannian sense.

2025

In this talk we first recall the notion of a subriemannian manifold M and we provide various examples. Under some additional assumptions it is known that a subriemannian structure induces a hypo-elliptic non-negative operatorsub which is called sub-Laplacian. In the cases where M is a sphere (of a certain dimension) or a general compact two step nilmanifold we study the heat kernel and the spectrum ofsub . The results are compared with the heat kernel and the spectrum of the Laplacian on M. Similar to the case of Riemannian manifolds to which the Beltrami-Laplace operator is assigned to, one can study the relation between the subriemannian geometry of M and spectral invariants with respect tosub . This presentation is based on a joint

2025

In this paper we are interested in polynomial interpolation of irregular functions namely those elements of L 2 (R, µ) for µ a given probability measure. This is of course doesn't make any sense unless for L 2 functions that, at least, admit a continuous version. To characterize those functions we have, first, constructed, in an abstract fashion, a chain of Sobolev like subspaces of a given Hilbert space H 0 . Then we have proved that the chain of Sobolev like subspaces controls the existence of a continuous version for L 2 functions and gives a pointwise polynomial approximation with a quite accurate error estimation.

2025, Physical review

In the present paper we numerically construct new charged anti-de Sitter black holes coupled to nonlinear Born-Infeld electrodynamics within a certain class of scalar-tensor theories. The properties of the solutions are investigated both numerically and analytically. We also study the thermodynamics of the black holes in the canonical ensemble. For large values of the Born-Infeld parameter and for a certain interval of the charge values we find the existence of a first-order phase transition between small and very large black holes. An unexpected result is that for a certain small charge subinterval two phase transitions have been observed, one of zeroth and one of first order. It is important to note that such phase transitions are also observed for pure Einstein-Born-Infeld-AdS black holes.

2025, Proceedings of the American Mathematical Society

Let G G be a finite group acting freely in a CW-complex Σ m \Sigma ^{m} which is a homotopy m m -dimensional sphere and let f : Σ m → Y f:\Sigma ^{m} \to Y be a map of Σ m \Sigma ^{m} to a finite k k -dimensional CW-complex Y Y . We show that if m ≥ | G | k m\geq \vert G\vert k , then f f has an ( H , G ) (H,G) -coincidence for some nontrivial subgroup H H of G G .

2025, Journal of Differential Equations

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2025, Annals of Mathematics

2025, Geometry & Topology

We discuss dense embeddings of surface groups and fully residually free groups in topological groups. We show that a compact topological group contains a nonabelian dense free group of finite rank if and only if it contains a dense surface group. Also, we obtain a characterization of those Lie groups which admit a dense faithfully embedded surface group. Similarly, we show that any connected semisimple Lie group contains a dense copy of any fully residually free group. 22E40; 20H10

2025, Proceedings Mathematical Sciences

Let M be a Hilbert module of holomorphic functions over a natural function algebra A (Ω), where Ω ⊆ C m is a bounded domain. Let M 0 ⊆ M be the submodule of functions vanishing to order k on a hypersurface Z ⊆ Ω. We describe a method, which in principle may be used, to construct a set of complete unitary invariants for quotient modules Q = M ⊖ M 0 . The invariants are given explicitly in the particular case of k = 2.

2025, Israel Journal of Mathematics

In this note, we show that quasi-free Hilbert modules R defined over the polydisk algebra satisfying a certain positivity condition, defined via the hereditary functional calculus, admit a unique minimal dilation (actually a co-extension) to the Hardy module over the polydisk. An explicit realization of the dilation space is given along with the isometric embedding of the module R in it. Some consequences of this basic fact are then explored in the case of several natural function algebras.

2025, Journal of Geometry and Physics

In this work we introduce the category of multiplicative sections of an LA-groupoid. We prove that these categories carry natural strict Lie 2-algebra structures, which are Morita invariant. As applications, we study the algebraic structure underlying multiplicative vector fields on a Lie groupoid and in particular vector fields on differentiable stacks. We also introduce the notion of geometric vector field on the quotient stack of a Lie groupoid, showing that the space of such vector fields is a Lie algebra. We describe the Lie algebra of geometric vector fields in several cases, including classifying stacks, quotient stacks of regular Lie groupoids and in particular orbifolds, and foliation groupoids.

2025, Comptes Rendus Mathematique

We study multiplicative Dirac structures on Lie groups. We show that the characteristic foliation of a multiplicative Dirac structure is given by the cosets of a normal Lie subgroup and, whenever this subgroup is closed, the leaf space inherits the structure of a Poisson-Lie group. We also describe multiplicative Dirac structures on Lie groups infinitesimally.

2025, arXiv: Commutative Algebra

Consider the ring of smooth function germs at the origin of mathbbRn\mathbb{R}^nmathbbRn. The Taylor expansion is the completion map from this ring to the ring of formal power series. Borel's lemma ensures the surjectivity of this map. In this short note we address the surjectivity of the completion map for the rings of smooth functions and rather general filtrations. This generalizes the classical Whitney extension theorem.

2025, Journal of Pure and Applied Algebra

Artin approximation and other related approximation results are used in various areas. The traditional formulation of such results is restricted to filtrations by powers of ideals, {I j }, and to Noetherian rings. In this paper we extend... more

2025, Journal of Pure and Applied Algebra

We consider matrices with entries in a local ring, M at m×n (R). Fix a group action, G M at m×n (R), and a subset of allowed deformations, Σ ⊆ M at m×n (R). The standard question in Singularity Theory is the finite-(Σ, G)-determinacy of matrices. Finite determinacy implies algebraizability and is equivalent to a stronger notion: stable algebraizability. In our previous work this determinacy question was reduced to the study of the tangent spaces T (Σ,A) , T (GA,A) , and their quotient, the tangent module to the miniversal deformation, T 1 (Σ,G,A) = T (Σ,A) T (GA,A) . In particular, the order of determinacy is controlled by the annihilator of this tangent module, ann(T 1 (Σ,G,A) ). In this work we study this tangent module for the group action GL(m, R)×GL(n, R) M at m×n (R) and various natural subgroups of it. We obtain ready-to-use criteria of determinacy for deformations of (embedded) modules, (skew-)symmetric forms, filtered modules, filtered morphisms of filtered modules, chains of modules and others.

2025, European Journal of Mathematics

We suggest the necessary/sufficient criteria for existence of a (order-byorder) solution y(x) of a functional equation F(x, y) = 0 over a ring. In full generality, the criteria hold in the category of filtered groups, this includes the wide class of modules over (commutative, associative) rings. The classical Implicit Function Theorem and its strengthening obtained by Tougeron and Fisher appear to be (weaker) particular forms of the general criterion. We obtain a special criterion for solvability of equations arising from group actions g(w) = w + u, here u is "small". As an immediate application we re-derive the classical criteria of determinacy, in terms of the tangent space to the orbit. Finally, we prove the Artin-Tougeron-type approximation theorem: if a system of C ∞ -equations has a formal solution and the derivative satisfies a Lojasiewicz-type condition then the system has a C ∞ -solution.

2025, Physical Review B

We consider a stack of parallel sheets composed of conducting planes with tensorial conductivities. Using the scattering matrix approach, we derive explicit formulas for the Casimir energy of two, three, and four planes, as well as a recurrence relation for arbitrary planes. Specifically, for a stack of graphene, we solve the recurrence relations and obtain formulas for the Casimir energy and force acting on the planes within the stack. Moreover, we calculate the binding energy in the graphene stack with graphite interplane separation, which amounts to E i b = 9.9 meV/atom. Notably, the Casimir force on graphene sheets decreases rapidly for planes beyond the first one. In particular, for the second graphene layer in the stack, the force is 35 times smaller than that experienced by the first layer.

2025, arXiv (Cornell University)

We use q-Pascal's triangle to define a family of representations of dimension 6 of the braid group B 3 on three strings. Then we give a necessary and sufficient condition for these representations to be irreducible.

2025, Modern Physics Letters A

Recent results show that when nonlinear electrodynamics is considered, the no-scalar-hair theorems in the scalar–tensor theories (STT) of gravity, which are valid for the cases of neutral black holes and charged black holes in the Maxwell electrodynamics, can be circumvented.1,2 What is even more, in the present work, we find new non-unique, numerical solutions describing charged black holes coupled to nonlinear electrodynamics in a special class of scalar–tensor theories. One of the phases has a trivial scalar field and coincides with the corresponding solution in General Relativity. The other phases that we find are characterized by the value of the scalar field charge. The causal structure and some aspects of the stability of the solutions have also been studied. For the scalar–tensor theories considered, the black holes have a single, non-degenerate horizon, i.e. their causal structure resembles that of the Schwarzschild black hole. The thermodynamic analysis of the stability of...

2025, Georgian Mathematical Journal

This work presents a general framework that innovates and explores different mathematical aspects associated with special functions by utilizing the mathematical physics-based idea of monomiality. This study presents a unique family of multivariable Hermite polynomials that are closely related to Frobenius-Genocchi polynomials of Apostol type. The study's deductions address the differential equation, generating expression, operational formalism, and other characteristics that define these polynomials. The affirmation of the controlling monomiality principle further confirms their mathematical foundations. In addition, the work proves recurrence relations, fractional operators, summation formulae, series representations, operational and symmetric identities, and so on, all of which contribute to our knowledge of these complex polynomials.

2025

We explore a new (simpler) formulation of the electromagnetic 4-force. The result is shown to be expressible using the 4D vector product of ref. [8] (ref. [8] is attached for reference). The electromagnetic 4-force is shown to be expressible as the 4D cross product of the 4-velocity with a 4-vector function Y which combines the E and H field. Maxwell's equations are also expressible in terms of Y.

2025

We compute the asymptotic number of monic trace-one integral polynomials with Galois group C3 and bounded height. For such polynomials we compute a height function coming from toric geometry and introduce a parametrization using the quadratic cyclotomic field Q(√-3). We also give a formula for the number of polynomials of the form t 3-t 2 + at + b ∈ Z[t] with Galois group C3 for a fixed integer a.

2025, Ann. Acad. Sci. Fenn. Math

The end of a hyperbolic surface is studied in terms of the behavior at infinity of geodesics on the surface. For a class of surfaces called untwisted flutes it is possible to give a fairly precise description of the ending geometry. From the point of view of a Fuchsian group representing such a surface this provides new information about the existence of Dirichlet and Garnett points.

2025

2025, Mathematics

In recent years, various methods and directions for solving a system of Boolean algebraic equations have been invented, and now they are being very actively investigated. One of these directions is the method of transforming a system of Boolean algebraic equations, given over a ring of Boolean polynomials, into systems of equations over a field of real numbers, and various optimization methods can be applied to these systems. In this paper, we propose a new transformation method for Solving Systems of Boolean Algebraic Equations (SBAE). The essence of the proposed method is that firstly, SBAE written with logical operations are transformed (approximated) in a system of harmonic-polynomial equations in the unit n-dimensional cube K n with the usual operations of addition and multiplication of numbers. Secondly, a transformed (approximated) system in K n is solved by using the optimization method. We substantiated the correctness and the right to exist of the proposed method with reliable evidence. Based on this work, plans for further research to improve the proposed method are outlined.

2025, Mathematics

In recent years, various methods and directions for solving a system of Boolean algebraic equations have been invented, and now they are being very actively investigated. One of these directions is the method of transforming a system of Boolean algebraic equations, given over a ring of Boolean polynomials, into systems of equations over a field of real numbers, and various optimization methods can be applied to these systems. In this paper, we propose a new transformation method for Solving Systems of Boolean Algebraic Equations (SBAE). The essence of the proposed method is that firstly, SBAE written with logical operations are transformed (approximated) in a system of harmonic-polynomial equations in the unit n-dimensional cube Kn with the usual operations of addition and multiplication of numbers. Secondly, a transformed (approximated) system in Kn is solved by using the optimization method. We substantiated the correctness and the right to exist of the proposed method with reli...

2025

We develop the Computational Least Action Principle (CLAP)-a variational framework in which the dynamical evolution of discrete algorithmic universes is selected by minimising a suitable information-theoretic action functional. After introducing computational configuration space, rule-space dynamics and Kolmogorov divergences, we prove that almost all trajectories inside the basin of an attractor render the computational action stationary under bootstrap-consistent variations. In the continuum limit the formalism reproduces both the Einstein-Hilbert action of General Relativity and the standard local action of quantum field theory, thereby providing a unifying information-centric explanation of known physical laws. Planck-scale corrections, computational decoherence rates and algorithmic predictions for conscious systems are discussed.

2025

In this article, we examine the Perturbed Gerdjikov-Ivanov equation by using the Jacobi ellipticfunction expansion method. This results in obtaining distinct solutions including dark, bright,singular solitons, periodic waves, singular periodic waves, and Jacobi elliptic function solutions. The2- and 3-dimensional graphs of the reported solutions are presented. The reported results may beuseful in explaining the physical features of the studied equation.

2025, arXiv (Cornell University)

The classification of the long-term behavior of dynamical systems is a fundamental problem in mathematics. For both deterministic and stochastic dynamics specific classes of models verify Palis' conjecture: the long-term behavior is determined by a finite number of stationary distributions. In this paper we consider the classification problem for stochastic models of interacting species. For a large class of three-species, stochastic differential equation models, we prove a variant of Palis' conjecture: the long-term statistical behavior is determined by a finite number of stationary distributions and, generically, three general types of behavior are possible: 1) convergence to a unique stationary distribution that supports all species, 2) convergence to one of a finite number of stationary distributions supporting two or fewer species, 3) convergence to convex combinations of single species, stationary distributions due to a rock-paper-scissors type of dynamic. Moreover, we prove that the classification reduces to computing Lyapunov exponents (external Lyapunov exponents) that correspond to the average per-capita growth rate of species when rare. Our results stand in contrast to the deterministic setting where the classification is incomplete even for three-dimensional, competitive Lotka-Volterra systems. For these SDE models, our results also provide a rigorous foundation for ecology's modern coexistence theory (MCT) which assumes the external Lyapunov exponents determine long-term ecological outcomes.

2025, Progress in Mathematics

A recent notion in theoretical physics is that not all quantum theories arise from quantising a classical system. Also, a given quantum model may possess more than just one classical limit. These facts find strong evidence in string duality and M-theory, and it has been suggested that they should also have a counterpart in quantum mechanics. In view of these developments we propose dequantisation, a mechanism to render a quantum theory classical. Specifically, we present a geometric procedure to dequantise a given quantum mechanics (regardless of its classical origin, if any) to possibly different classical limits, whose quantisation gives back the original quantum theory. The standard classical limit h → 0 arises as a particular case of our approach.

2025, Tokyo Journal of Mathematics

Throughout this paper, we use mainly the same notations as in [8].

2025, J. Lie Theory

The quantum picture is basically set up by the Weyl algebra. It is derived from the differential calculus via correspondence principle: Let u be the operator x· of multiplication by the coordinate function x on R acting on the space of... more

2025, Advances in Mathematical Physics

2025, SIAM Journal on Mathematics of Data Science

Data sets sampled in Lie groups are widespread, and as with multivariate data, it is important for many applications to assess the differences between the sets in terms of their distributions. Indices for this task are usually derived by considering the Lie group as a Riemannian manifold. Then, however, compatibility with the group operation is guaranteed only if a bi-invariant metric exists, which is not the case for most non-compact and non-commutative groups. We show here that if one considers an affine connection structure instead, one obtains bi-invariant generalizations of well-known dissimilarity measures: a Hotelling T 2 statistic, Bhattacharyya distance and Hellinger distance. Each of the dissimilarity measures matches its multivariate counterpart for Euclidean data and is translation-invariant, so that biases, e.g., through an arbitrary choice of reference, are avoided. We further derive non-parametric two-sample tests that are bi-invariant and consistent. We demonstrate the potential of these dissimilarity measures by performing group tests on data of knee configurations and epidemiological shape data. Significant differences are revealed in both cases.

2025, Canadian Mathematical Bulletin

For a parabolic subgroup H of the general linear group G = Gl(n, C), we characterize the Kähler classes of G/H and give a formula for the height of any two-dimensional cohomology class. As an application, we classify the automorphisms of... more

2025, Springer eBooks

2025, Physics Letters B

We analyze the exact perturbative solution of N = 2 Born-Infeld theory which is believed to be defined by Ketov's equation. This equation can be considered as a truncation of an infinite system of coupled differential equations defining Born-Infeld action with one manifest N = 2 and one hidden N = 2 supersymmetries. We explicitly demonstrate that infinitely many new structures appear in the higher orders of the perturbative solution to Ketov's equation. Thus, the full solution cannot be represented as a function depending on a finite number of its arguments. We propose a mechanism for generating the new structures in the solution and show how it works up to 18-th order. Finally, we discuss two new superfield actions containing an infinite number of terms and sharing some common features with N = 2 supersymmetric Born-Infeld action.

2025, Banach Journal of Mathematical Analysis

In this short note, a new approach is provided to prove that every nonzero continuous cosine function on a compact group G is the normalized character of a representation of G into the special unitary group SU (2).

2025, Advances in Operator Theory

In this paper, the relations between the Yang-Baxter equation and affine actions are explored in detail. In particular, we classify solutions of the Yang-Baxter equations in two ways: (i) by their associated affine actions of their structure groups on their derived structure groups, and (ii) by the C*-dynamical systems obtained from their associated affine actions. On the way to our main results, several other useful results are also obtained. 2010 Mathematics Subject Classification. 16T25.

2025, Banach Journal of Mathematical Analysis

2025, Israel Journal of Mathematics

Let G be a finitely generated pro-p group, equipped with the ppower series P : G i = G p i , i ∈ N 0 . The associated metric and Hausdorff dimension function hdim P the Hausdorff spectrum of closed subgroups of G. In the case where G is p-adic analytic, the Hausdorff dimension function is well understood; in particular, hspec P (G) consists of finitely many rational numbers closely linked to the analytic dimensions of subgroups of G. Conversely, it is a long-standing open question whether |hspec P (G)| < ∞ implies that G is p-adic analytic. We prove that the answer is yes, in a strong sense, under the extra condition that G is soluble. Furthermore, we explore the problem and related questions also for other filtration series, such as the lower p-series, the Frattini series, the modular dimension subgroup series and quite general filtration series. For instance, we prove, for p > 2, that every countably based pro-p group G with an open subgroup mapping onto Z p ⊕ Z p admits a filtration series S such that hspec S (G) contains an infinite real interval.

2025, International Journal of Modern Physics A

We will present the axioms of Bogoliubov’s causal perturbative QFT in which the creation-annihilation operators are interpreted as Hida operators. We will shortly present the results that can be achieved in this theory: (1) removal of UV and IR infinity in the scattering operator, (2) existence of the adiabatic limit for interacting fields in QED, (3) proof that charged particles have nonzero mass, (4) existence of infrared and ultraviolet asymptotics for QED.

2025, Journal of physics

The internal space for a molecule, atom, or other n-body system can be conveniently parameterised by 3n -9 kinematic angles and three kinematic invariants. For a fixed set of kinematic invariants, the kinematic angles parameterise a subspace, called a kinematic orbit, of the n-body internal space. Building on an earlier analysis of the three-and four-body problems, we derive the form of these kinematic orbits (that is, their topology) for the general nbody problem. The case n = 5 is studied in detail, along with the previously studied cases n = 3, 4.

2025, arXiv (Cornell University)

In this paper we study the theories of the infinite-branching tree and the r-regular tree, and show that both of them are pseudofinite. Moreover, we show that they can be realized by infinite ultraproducts of polynomial exact classes of... more

2025

Families of exact solutions are found to a nonlinear modification of the Black-Scholes equation. This risk-adjusted pricing methodology model (RAPM) incorporates both transaction costs and the risk from a volatile portfolio. Using the Lie group analysis we obtain the Lie algebra admitted by the RAPM equation. It gives us the possibility to describe an optimal system of subalgebras and the corresponding set of invariant solutions to the model. In this way we can describe the complete set of possible reductions of the nonlinear RAPM model. Reductions are given in the form of di↵erent second order ordinary di↵erential equations. In all cases we provide exact solutions to these equations in an explicit or parametric form. Each of these solutions contains a reasonable set of parameters which allows one to approximate a wide class of boundary conditions. We discuss the properties of these reductions and the corresponding invariant solutions.