Algebraic Field Research Articles

Formula omitted] modules of finite type on affine space

We study the category of modules admitting compatible actions of the Lie algebra V of vector fields on an affine space and the algebra A of polynomial functions. We show that modules in this category which are finitely generated over A, are free. We also show that this pair of compatible actions is equivalent to commuting actions of the algebra of differential operators and the Lie algebra of vector fields vanishing at the origin. This allows us to construct explicit realizations of such modules as gauge modules.

Confinement in non-Abelian lattice gauge theory via persistent homology

We investigate the structure of confining and deconfining phases in SU(2) lattice gauge theory via persistent homology, which gives us access to the topology of a hierarchy of combinatorial objects constructed from given data. Specifically, we use filtrations by traced Polyakov loops, topological densities, holonomy Lie algebra fields, as well as electric and magnetic fields. This allows for a comprehensive picture of confinement. In particular, topological densities form spatial lumps which show signatures of the classical probability distribution of instanton-dyons. Signatures of well-separated dyons located at random positions are encoded in holonomy Lie algebra fields, following the semiclassical temperature dependence of the instanton appearance probability. Debye screening discriminating between electric and magnetic fields is visible in persistent homology and pronounced at large gauge coupling. All employed constructions are gauge-invariant without a priori assumptions on the configurations under study. This work showcases the versatility of persistent homology for statistical and quantum physics studies, barely explored to date.

Exploration of the distillability problem

Quantum mechanics has a history more than 100 years, and there are still many challenging open-ended questions. The problems encountered in modern physics such as matrix mechanics and quantum information often involve both quantum theory and open-ended problems in modern mathematics such as linear algebra. Distillability problem is an important open-ended problem in the field of quantum information and linear algebra. This article recalls the development history and application of linear algebra, summarizes the background and progress of distillability conjecture, and attempts to explore this problem by summarizing and using the knowledge of linear algebra and matrix theory learned in the course.

Cosimplicial meromorphic functions cohomology on complex manifolds

Developing ideas of [B. L. Feigin, Conformal field theory and cohomologies of the Lie algebra of holomorphic vector fields on a complex curve, in Proc. Int. Congress of Mathematicians (Kyoto, 1990), Vols. 1 and 2 (Mathematical Society of Japan, Tokyo, 1991), pp. 71–85], we introduce canonical cosimplicial cohomology of meromorphic functions for infinite-dimensional Lie algebra formal series with prescribed analytic behavior on domains of a complex manifold [Formula: see text]. Graded differential cohomology of a sheaf of Lie algebras [Formula: see text] via the cosimplicial cohomology of [Formula: see text]-formal series for any covering by Stein spaces on [Formula: see text] is computed. A relation between cosimplicial cohomology (on a special set of open domains of [Formula: see text]) of formal series of an infinite-dimensional Lie algebra [Formula: see text] and singular cohomology of auxiliary manifold associated to a [Formula: see text]-module is found. Finally, multiple applications in conformal field theory, deformation theory, and in the theory of foliations are proposed.

Noncommutative Correction to the Entropy of Charged BTZ Black Hole

Noncommutative geometry is an established potential candidate for including quantum phenomena in gravitation. We outlined the formalism of Hopf algebras and its connection to the algebra of infinitesimal diffeomorphisms. Using a Drinfeld twist, we deformed spacetime symmetries, algebra of vector fields and differential forms, leading to a formulation of noncommutative Einstein equations. We studied a concrete example of charged BTZ spacetime and deformations steaming from the so-called angular twist. The entropy of the noncommutative charged BTZ black hole was obtained using the brick-wall method. We used a charged scalar field as a probe and obtained its spectrum and density of states via WKB approximation. We provide the method used to calculate corrections to the Bekenstein–Hawking entropy in higher orders in WKB, but we present the final result in the lowest WKB order. The result is that, even in the lowest order in WKB, the entropy, in general, contains higher powers in ℏ, and it has logarithmic corrections and polynomials of logarithms of the black hole area.

Cosets of Free Field Algebras via Arc Spaces

AbstractUsing the invariant theory of arc spaces, we find minimal strong generating sets for certain cosets of affine vertex algebras inside free field algebras that are related to classical Howe duality. These results have several applications. First, for any vertex algebra ${{\mathcal {V}}}$, we have a surjective homomorphism of differential algebras $\mathbb {C}[J_{\infty }(X_{{{\mathcal {V}}}})] \rightarrow \text {gr}^{F}({{\mathcal {V}}})$; equivalently, the singular support of ${{\mathcal {V}}}$ is a closed subscheme of the arc space of the associated scheme $X_{{{\mathcal {V}}}}$. We give many new examples of classically free vertex algebras (i.e., this map is an isomorphism), including $L_{k}({{\mathfrak {s}}}{{\mathfrak {p}}}_{2n})$ for all positive integers $n$ and $k$. We also give new examples where the kernel of this map is nontrivial but is finitely generated as a differential ideal. Next, we prove a coset realization of the subregular ${{\mathcal {W}}}$-algebra of ${{\mathfrak {s}}}{{\mathfrak {l}}}_{n}$ at a critical level that was previously conjectured by Creutzig, Gao, and the 1st author. Finally, we give some new level-rank dualities involving affine vertex superalgebras.

Algebraic dimension and complex subvarieties of hypercomplex nilmanifolds

A nilmanifold is a (left) quotient of a nilpotent Lie group by a cocompact lattice. A hypercomplex structure on a manifold is a triple of complex structure operators satisfying the quaternionic relations. A hypercomplex nilmanifold is a compact quotient of a nilpotent Lie group equipped with a left-invariant hypercomplex structure. Such a manifold admits a whole 2-dimensional sphere S2 of complex structures induced by quaternions. We prove that for any hypercomplex nilmanifold M and a generic complex structure L∈S2, the complex manifold (M,L) has algebraic dimension 0. A stronger result is proven when the hypercomplex nilmanifold is abelian. Consider the Lie algebra of left-invariant vector fields of Hodge type (1,0) on the corresponding nilpotent Lie group with respect to some complex structure I∈S2. A hypercomplex nilmanifold is called abelian when this Lie algebra is abelian. We prove that all complex subvarieties of (M,L) for generic L∈S2 on a hypercomplex abelian nilmanifold are also hypercomplex nilmanifolds.

Левоинвариантная парасасакиева структура на групповой модели вещественного расширения плоскости де-Ситтера

In this paper, a group model for a real extension of the de Sitter plane is pro-posed. This group contains a group of special matrices, which is a subgroup of the general linear group. It is established that there exists a left-invariant contact metric structure on this group, which is normal and, therefore, para-Sasakian. The basis vector fields of the Lie algebra of infinitesimal automorphisms are found. The Lie group of automorphisms has the maximum dimension and, in addition to the Levi-Civita connection, it also retains a contact metric connection with skew-symmetric torsion. In this connection, all structural tensors of the para-Sasakian structure, as well as the torsion and curvature tensors, are covariantly constant. Using a nonholonomic field of orthonormal frames adapted to the contact distribution, an orthogonal projection of the Levi-Civita connection onto the contact distribution is found, which is a truncated connection. Passing to natural coordinates, differential equations of geodesics of the truncated connection and Levi-Civita connection are found. Thus, the Levi-Civita contact geodesic connections coincide with the truncated connection geodesics. This means that through each point in each contact direction there is a unique Levi-Civita geodesic connection tangent to the contact distribution. The Levi-Civita connection, like the contact metric connection, is consistent with the contact distribution.

On conjunctive complex fuzzification of Lagrange's theorem of <i>ξ</i>−CFSG

<abstract><p>The application of a complex fuzzy logic system based on a linear conjunctive operator represents a significant advancement in the field of data analysis and modeling, particularly for studying physical scenarios with multiple options. This approach is highly effective in situations where the data involved is complex, imprecise and uncertain. The linear conjunctive operator is a key component of the fuzzy logic system used in this method. This operator allows for the combination of multiple input variables in a systematic way, generating a rule base that captures the behavior of the system being studied. The effectiveness of this method is particularly notable in the study of phenomena in the actual world that exhibit periodic behavior. The foremost aim of this paper is to contribute to the field of fuzzy algebra by introducing and exploring new concepts and their properties in the context of conjunctive complex fuzzy environment. In this paper, the conjunctive complex fuzzy order of an element belonging to a conjunctive complex fuzzy subgroup of a finite group is introduced. Several algebraic properties of this concept are established and a formula is developed to calculate the conjunctive complex fuzzy order of any of its powers in this study. Moreover, an important condition is investigated that determines the relationship between the membership values of any two elements and the membership value of the identity element in the conjunctive complex fuzzy subgroup of a group. In addition, the concepts of the conjunctive complex fuzzy order and index of a conjunctive complex fuzzy subgroup of a group are also presented in this article and their various fundamental algebraic attributes are explored structural. Finally, the conjunctive complex fuzzification of Lagrange's theorem for conjunctive complex fuzzy subgroups of a group is demonstrated.</p></abstract>

Higher-Order Associativity in Field Algebras

Field algebras were defined by Bakalov and Kac as an associative analogue of vertex algebras. We define the notion of higher-order associativity for field algebras and construct examples to show that higher-order associativity imposes a strictly stronger condition on field algebras than lower-order associativity.

A New Perspective on $k$-Ideals of a Semiring via Soft Intersection Ideals

In recent years, soft sets have become popular in various fields. For this reason, many studies have been carried out in the field of algebra. In this study, soft intersection k-ideals are defined with the help of a semiring, and some algebraic structures are examined. Moreover, the quotient rings are defined by k-semiring. Isomorphism theorems are examined by quotient rings. Finally, some algebraic properties are investigated by defining soft intersection maximal k-ideals.

Локально изометричные римановы аналитические пространства

Classes of locally isometric Riemannian analytic manifolds are studied. A generalization of the concept of completeness is given. We consider the Lie algebra 𝔤 of all Killing vector fields of a Riemannian analytic manifold, its stationary subalgebra 𝔥 the simply connected Lie group 𝐺 corresponding to the Lie algebra 𝔤, and the subgroup 𝐻 corresponding to the Lie subalgebra 𝔥. In the absence of a center in the algebra 𝔤 the concept of a quasi-complete (compressed) manifold is introduced. An oriented Riemannian analytic manifold whose vector field algebra has zero center is said to be quasi-complete if it is non-extendable and does not admit non-trivial orientation-preserving and all Killing vector fields local isometries to itself. The main property of such a manifold is that it is unique in the class of all locally isometric Riemannian analytic manifolds, and any locally given isometry of this manifold 𝑀 into itself can be analytically extended to an isometry 𝑓: 𝑀 ≈ 𝑀. For an arbitrary class of locally isometric Riemannian analytic manifolds, a definition of a pseudocomplete manifold is given, which is complete if a complete manifold exists in the given class. A Riemannian analytic simply connected manifold M is called pseudocomplete if it has the following properties. 𝑀 is non-extendable. There is no locally isometric covering map f; M→N, where N is a simply connected Riemannian analytic manifold and f (M) is an open subset of N not equal to N.

On generalized derivations of polynomial vector fields Lie algebras

In this paper, we study the generalized derivation of a Lie sub-algebra of the Lie algebra of polynomial vector fields on $\mathbb{R}^n$ where $n\geq1$, containing all constant vector fields and the Euler vector field, under some conditions on this Lie sub-algebra.

Applications of Group Theory

Groups play a fundamental role in Algebra: many algebraic structures, including rings, fields, and modules, can be seen as formed by adding new operations and axioms based on groups. Researchers often use group theory to explain many kinds of phenomena. In recent years, group theory has been introduced into crystallography to further explore the macroscopic symmetry of crystals from a mathematical point of view. In this paper, the applications of group theory in crystallography and magic cubic will be discussed. Basic definitions and models of these fields are demonstrated. A finite group is a group with a finite number of elements, which are the important contents of group theory. Besides, this paper proves that n−1 elements in a n order group can completely decide the nth element and gives a method of the nth element in a commutative group of order n. The analysis suggests that the research method of group theory has an important influence on other subjects.

Performance Comparison of Channel Coding Techniques for OFDM System

Wireless communication is one of the speedily evolving and dynamic technology areas in the realm of communication. OFDM is an acronym for Orthogonal Frequency Division Multiplexing. It is a version of FDM in which orthogonal subcarriers are closely spaced to maximize bandwidth consumption. By decreasing Inter Symbol Interference (ISI) and increasing the Signal-to-Noise ratio (SNR), it has enhanced the quality of long-distance communication. It’s commonly employed in digital communication systems to safeguard digital data from noise and interference while also reducing bit mistakes. Convolutional codes, Block codes, are the two fundamental forms of channel codes. Block codes are based on abstract algebra and finite field arithmetic. They are capable of detecting and correcting faults. This category includes LDPC and Reed Solomon codes. Convolution codes, alternatively are created using a distinct strong mathematical framework and are generally utilised for real-time error correction. They combine all of the data into a single codeword. Reed-Solomon codes are block-based error correcting codes used in digital communications, storage, and ADSL, among other uses. The BER performance of an OFDM system is investigated in this paper using various channel coding algorithms.

On Lie’s classification of subalgebras of vector fields on the plane

A brief proof of Lie’s classification of solvable algebras of vector fields on the plane is given. The proof uses basic representation theory and PDEs.

Splitting fields of differential symbol algebras

For m≥2, we study derivations on symbol algebras of degree m over fields with characteristic not dividing m. A differential central simple algebra over a field k is split by a finitely generated extension of k. For certain derivations on symbol algebras, we provide explicit construction of differential splitting fields and give bounds on their algebraic and transcendence degrees. We further analyze maximal subfields that split certain differential symbol algebras.

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