Algebraic Field Research Articles

Decomposable Five-Dimensional Lie Algebras in the Problem on Holomorphic Homogeneity in ℂ3

In connection with the problem of describing holomorphically homogeneous real hypersurfaces in the space ℂ3, we study five-dimensional real Lie algebras realized as algebras of holomorphic vector fields on such manifolds. We prove the following assertion: If on a holomorphically homogeneous real hypersurface M of the space ℂ3, there is a decomposable, solvable, five-dimensional Lie algebra of holomorphic vector fields having a full rank near some point P ∈ M, then this surface is either degenerate near P in the sense of Levy or is a holomorphic image of an affine-homogeneous surface.

Asymptotic Measurement Schemes for Every Observable of a Quantum Field Theory

In quantum measurement theory, a measurement scheme describes how an observable of a given system can be measured indirectly using a probe. The measurement scheme involves the specification of a probe theory, an initial probe state, a probe observable and a coupling between the system and the probe, so that a measurement of the probe observable after the coupling has ceased reproduces (in expectation) the result of measuring the system observable in the system state. Recent work has shown how local and causal measurement schemes may be described in the context of model-independent quantum field theory (QFT), but has not addressed the question of whether such measurement schemes exist for all system observables. Here, we present two treatments of this question. The first is a proof of principle which provides a measurement scheme for every local observable of the quantized real linear scalar field if one relaxes one of the conditions on a QFT measurement scheme by allowing a non-compact coupling region. Secondly, restricting to compact coupling regions, we explicitly construct asymptotic measurement schemes for every local observable of the quantized theory. More precisely, we show that for every local system observable A there is an associated collection of measurement schemes for system observables that converge to A. All the measurement schemes in this collection have the same fixed compact coupling zone and the same processing region. The convergence of the system observables holds, in particular, in GNS representations of suitable states on the field algebra or the Weyl algebra. In this way, we show that every observable can be asymptotically measured using locally coupled probe theories.

Classification of simple bounded weight modules of the Lie algebra of vector fields on ℂn

Classification of simple bounded weight modules of the Lie algebra of vector fields on ℂn

Infinitesimal automorphisms of algebraic varieties and vector fields on elliptic surfaces

We give several results concerning the connected component ${\rm Aut}_X^0$ of the automorphism scheme of a proper variety $X$ over a field, such as its behaviour with respect to birational modifications, normalization, restrictions to closed subschemes and deformations. Then, we apply our results to study the automorphism scheme of not necessarily Jacobian elliptic surfaces $f: X \to C$ over algebraically closed fields, generalizing work of Rudakov and Shafarevich, while giving counterexamples to some of their statements. We bound the dimension $h^0(X,T_X)$ of the space of global vector fields on an elliptic surface $X$ if the generic fiber of $f$ is ordinary or if $f$ admits no multiple fibers, and show that, without these assumptions, the number $h^0(X,T_X)$ can be arbitrarily large for any base curve $C$ and any field of positive characteristic. If $f$ is not isotrivial, we prove that ${\rm Aut}_X^0 \cong \mu_{p^n}$ and give a bound on $n$ in terms of the genus of $C$ and the multiplicity of multiple fibers of $f$. As a corollary, we re-prove the non-existence of global vector fields on K3 surfaces and calculate the connected component of the automorphism scheme of a generic supersingular Enriques surface in characteristic $2$. Finally, we present additional results on horizontal and vertical group scheme actions on elliptic surfaces which can be applied to determine ${\rm Aut}_X^0$ explicitly in many concrete cases.

Simple weight modules with finite weight multiplicities over the Lie algebra of polynomial vector fields

AbstractLet𝒲n{{\mathcal{W}}_{n}}be the Lie algebra of polynomial vector fields. We classify simple weight𝒲n{{\mathcal{W}}_{n}}-modulesMwith finite weight multiplicities. We prove that every such nontrivial moduleMis either a tensor module or the unique simple submodule in a tensor module associated with the de Rham complex onℂn{\mathbb{C}^{n}}.

Orthogonal Generalized Symmetric Higher Bi-Left (Resp. Right) Centralizers on Semiprime Γ- Ring

Specialists in the field of algebra have highlighted many results related to the Γ-ring, and in my research, I will present some definitions and concepts and a major theorem parallel to the results of other researchers.

Classification of the Realizations of Lie Algebras for the Vector Fields on a Circle

We describe the realizations of finite-dimensional Lie algebras of smooth tangential vector fields on a circle and construct “canonical” realizations of the two-dimensional noncommutative algebra, as well as the algebra 𝔰𝔩(2,ℝ) . It is shown that any realization of these algebras by smooth vector fields can be reduced to one of “canonical” realizations with the help of piecewise-smooth global transformations of a circle onto itself. We also deduce the formulas for the number of nonequivalent realizations.

CR embeddings of CR manifolds

We improve results of Baouendi, Rothschild and Treves and of Hill and Nacinovich by finding a much weaker sufficient condition for a CR manifold of type (n, k) to admit a local CR embedding into a CR manifold of type (n+ell ,k-ell ). While their results require the existence of a finite dimensional solvable transverse Lie algebra of vector fields, we require only a finite dimensional extension.

Dimensionality reduction of longitudinal 'omics data using modern tensor factorizations.

Longitudinal 'omics analytical methods are extensively used in the evolving field of precision medicine, by enabling 'big data' recording and high-resolution interpretation of complex datasets, driven by individual variations in response to perturbations such as disease pathogenesis, medical treatment or changes in lifestyle. However, inherent technical limitations in biomedical studies often result in the generation of feature-rich and sample-limited datasets. Analyzing such data using conventional modalities often proves to be challenging since the repeated, high-dimensional measurements overload the outlook with inconsequential variations that must be filtered from the data in order to find the true, biologically relevant signal. Tensor methods for the analysis and meaningful representation of multiway data may prove useful to the biological research community by their advertised ability to tackle this challenge. In this study, we present tcam-a new unsupervised tensor factorization method for the analysis of multiway data. Building on top of cutting-edge developments in the field of tensor-tensor algebra, we characterize the unique mathematical properties of our method, namely, 1) preservation of geometric and statistical traits of the data, which enable uncovering information beyond the inter-individual variation that often takes over the focus, especially in human studies. 2) Natural and straightforward out-of-sample extension, making tcam amenable for integration in machine learning workflows. A series of re-analyses of real-world, human experimental datasets showcase these theoretical properties, while providing empirical confirmation of tcam's utility in the analysis of longitudinal 'omics data.

Reduction and reconstruction of multisymplectic Lie systems

A Lie system is a non-autonomous system of first-order ordinary differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional real Lie algebra of vector fields, a so-called Vessiot–Guldberg Lie algebra. In this work, multisymplectic forms are applied to the study of the reduction of Lie systems through their Lie symmetries. By using a momentum map, we perform a reduction and reconstruction procedure of multisymplectic Lie systems, which allows us to solve the original problem by analysing several simpler multisymplectic Lie systems. Conversely, we study how reduced multisymplectic Lie systems allow us to retrieve the form of the multisymplectic Lie system that gave rise to them. Our results are illustrated with examples from physics, mathematics, and control theory.

Baby universes and worldline field theories

The quantum gravity path integral involves a sum over topologies that invites comparisons to worldsheet string theory and to Feynman diagrams of quantum field theory. However, the latter are naturally associated with the non-abelian algebra of quantum fields, while the former has been argued to define an abelian algebra of superselected observables associated with partition-function-like quantities at an asymptotic boundary. We resolve this apparent tension by pointing out a variety of discrete choices that must be made in constructing a Hilbert space from such path integrals, and arguing that the natural choices for quantum gravity differ from those used to construct QFTs. We focus on one-dimensional models of quantum gravity in order to make direct comparisons with worldline QFT. We also restrict attention to models in which worldlines do not split apart or join together.

S -wave pion condensation in symmetric nuclear matter

S-wave pion-nucleon interactions in the linear sigma model, and in Manohar-Georgi and Gasser-Sainio-Svarc models with finite number of terms in Lagrangians, as well as in a general phenomenological approach are reviewed. Subtleties associated with the current algebra theorems and field redefinitions are discussed. In the first and third models most likely the s-wave pion condensation in the isospin-symmetric matter does not occur at least up to high densities, whereas within the second model it may appear already at moderate densities. In the phenomenological approach two parameterizations of the s-wave pion-nucleon scattering amplitude and the pion polarization operator used in the literature are considered. The first parameterization employs the off-mass-shell amplitude and allows to fulfil the current algebra theorems. Using it the s-wave pion polarization operator in the isospin-symmetric matter is reconstructed within the gas approximation. With this pion polarization operator the s-wave pion condensation in the isospin-symmetric matter does not occur at least up to high densities. Second parameterization uses the on-mass-shell pion-nucleon scattering amplitude and does not satisfy the Adler and Weinberg conditions. With such a parameterization most likely the s-wave pion condensation in the isospin-symmetric matter may occur already at the nucleon density $n\simeq (1.4-2.5) n_0$, where $n_0$ is the density of the atomic nucleus, that should result in observable effects. Both parameterizations allow to successfully describe the pion atom data.

Common splitting fields of symbol algebras

We study the common splitting fields of symbol algebras of degree \(p^m\) over fields F of \({\text {char}}(F)=p\). We first show that if any finite number of such algebras share a degree \(p^m\) simple purely inseparable splitting field, then they share a cyclic splitting field of the same degree. As a consequence, we conclude that every finite number of symbol algebras of degrees \(p^{m_0},\dots ,p^{m_t}\) share a cyclic splitting field of degree \(p^{m_0+\dots +m_t}\). This generalization recovers the known fact that every tensor product of symbol algebras is a symbol algebra. We apply a result of Tignol’s to bound the symbol length of classes in \({\text {Br}}_{p^m}(F)\) whose symbol length when embedded into \({\text {Br}}_{p^{m+1}}(F)\) is 2 for \(p\in \{2,3\}\). We also study similar situations in other Kato-Milne cohomology groups, where the necessary norm conditions for splitting exist.

Uncertainties in quantum measurements: a quantum tomography

The observables associated with a quantum system S form a non-commutative algebra . It is assumed that a density matrix ρ can be determined from the expectation values of observables. But admits inner automorphisms , , so that its individual elements can be identified only up to unitary transformations. So since Tr ρ(uau*) = Tr(u*ρu)a, only the spectrum of ρ, or its characteristic polynomial, can be determined in quantum mechanics. In local quantum field theory, ρ cannot be determined at all, as we shall explain. However, abelian algebras do not have inner automorphisms, so the measurement apparatus can determine mean values of observables in abelian algebras (M for measurement, S for system). We study the uncertainties in extending to (the determination of which means measurement of ) and devise a protocol to determine by determining for different choices of . The problem we formulate and study is a generalization of the Kadison–Singer theorem. We give an example where the system S is a particle on a circle and the experiment measures the abelian algebra of a magnetic field B coupled to S. The measurement of B gives information about the state ρ of the system S due to operator mixing. Associated uncertainty principles for von Neumann entropy are discussed in the appendix, adapting the earlier work by Białynicki-Birula and Mycielski (1975 Commun. Math. Phys. 44 129) to the present case.

Whittaker category for the Lie algebra of polynomial vector fields

For any positive integer n, let An=C[t1,…,tn], Wn=Der(An) and Δn=Span{∂∂t1,…,∂∂tn}. Then (Wn,Δn) is a Whittaker pair. A Wn-module M is called a Whittaker module if the action of Δn on M is locally finite. We show that each block ΩaW˜ of the category of (An,Wn)-Whittaker modules with finite dimensional Whittaker vector spaces is equivalent to the category of finite dimensional modules over Ln, where Ln is the Lie subalgebra of Wn consisting of vector fields vanishing at the origin. As a corollary, we classify all simple non-singular Whittaker Wn-modules with finite dimensional Whittaker vector spaces using gln-modules. We also obtain an analogue of Skryabin's equivalence for the non-singular block ΩaW.

Lie Algebra of Killing Vector Fields and its Stationary Subalgebra

Let 𝔤 be the Lie algebra of all Killing vector fields on a locally homogeneous, analytic Riemannian manifold M, 𝔥 be a stationary subalgebra of 𝔤, G be the simply connected group generated by the algebra 𝔤, H be the subgroup of G generated by the subalgebra 𝔥, 𝔷 be the center of the algebra g, r be its radical, and [𝔤; 𝔤] be its commutator subgroup. If dim (𝔥 ∩ (𝔷 + [𝔤, 𝔤])) = dim (𝔥 ∩ [𝔤, 𝔤]), then H is closed in G. If for any semisimple subalgebra 𝔭 ⊂ 𝔤 satisfying the condition 𝔭 + 𝔯 = 𝔤, the relation (𝔭 + 𝔷) ∩ 𝔥 = 𝔭 ∩ 𝔥 holds, then H is closed in G. We also examine the analytic continuation of the given local, analytic Riemannian manifold.

Hamiltonian structures of isomonodromic deformations on moduli spaces of parabolic connections

In this paper, we treat moduli spaces of parabolic connections. We take an affine open covering of the moduli spaces, and we construct a Hamiltonian structure of an algebraic vector field determined by the isomonodromic deformation for each affine open subset.

Ergodic Exploration Using Tensor Train: Applications in Insertion Tasks

In robotics, ergodic control extends the tracking principle by specifying a probability distribution over an area to cover instead of a trajectory to track. The original problem is formulated as a spectral multiscale coverage problem, typically requiring the spatial distribution to be decomposed as Fourier series. This approach does not scale well to control problems requiring exploration in search space of more than two dimensions. To address this issue, we propose the use of tensor trains, a recent low-rank tensor decomposition technique from the field of multilinear algebra. The proposed solution is efficient, both computationally and storagewise, hence making it suitable for its online implementation in robotic systems. The approach is applied to a peg-in-hole insertion task requiring full 6-D end-effector poses, implemented with a seven-axis Franka Emika Panda robot. In this experiment, ergodic exploration allows the task to be achieved without requiring the use of force/torque sensors.

Strict deformation quantization of abelian lattice gauge fields

This paper shows how to construct classical and quantum field C*-algebras modeling a $U(1)^n$-gauge theory in any dimension using a novel approach to lattice gauge theory, while simultaneously constructing a strict deformation quantization between the respective field algebras. The construction starts with quantization maps defined on operator systems (instead of C*-algebras) associated to the lattices, in a way that quantization commutes with all lattice refinements, therefore giving rise to a quantization map on the continuum (meaning ultraviolet and infrared) limit. Although working with operator systems at the finite level, in the continuum limit we obtain genuine C*-algebras. We also prove that the C*-algebras (classical and quantum) are invariant under time evolutions related to the electric part of abelian Yang--Mills. Our classical and quantum systems at the finite level are essentially the ones of [van Nuland and Stienstra, 2020], which admit completely general dynamics, and we briefly discuss ways to extend this powerful result to the continuum limit. We also briefly discuss reduction, and how the current set-up should be generalized to the non-abelian case.

Всегда в моей памяти…

The article publishes memories of a mentor, an outstanding mathematician, academician Tashmukhamad Alievich Sarimsakov, who for many years conducted scientific activities at the Tashkent State University and was engaged in scientific development of “Theory of semi fields and its applications in probability theory”. He always supported new directions in science and gave his students the freedom to choose the direction of research, and then he himself connected to these studies, feeling their promise. At one time, the academician put forward the idea of creating a non-commutative analogue of semi fields for an axiomatic description of the algebra of random variables in non-commutative quantum probability theory. Sarimsakov also wrote several scientific monographs, one of them «Introduction to Quantum Probability Theory», which was highly appreciated by specialists. Several of his monographs and textbooks in the field of operator algebra were republished with additions, and he prepared a number of high-class specialists.