Two-dimensional Algebra Research Articles

Ternary Associativity and Ternary Lie Algebras at Cube Roots of Unity

We propose a new approach to extend the notion of commutator and Lie algebra to algebras with ternary multiplication laws. Our approach is based on the ternary associativity of the first and second kinds. We propose a ternary commutator, which is a linear combination of six triple products (all permutations of three elements). The coefficients of this linear combination are the cube roots of unity. We find an identity for the ternary commutator that holds due to the ternary associativity of either the first or second kind. The form of this identity is determined by the permutations of the general affine group GA(1,5)⊂S5. We consider this identity as a ternary analog of the Jacobi identity. Based on the results obtained, we introduce the concept of a ternary Lie algebra at cube roots of unity and provide examples of such algebras constructed using ternary multiplications of rectangular and three-dimensional matrices. We also highlight the connection between the structure constants of a ternary Lie algebra with three generators and an irreducible representation of the rotation group. The classification of two-dimensional ternary Lie algebras at cube roots of unity is proposed.

Group coactions on two-dimensional Artin-Schelter regular algebras

We describe all possible coactions of finite groups (equivalently, all group gradings) on two-dimensional Artin-Schelter regular algebras. We give necessary and sufficient conditions for the associated Auslander map to be an isomorphism, and determine precisely when the invariant ring for the coaction is Artin-Schelter regular. The proofs of our results are combinatorial and exploit the structure of the McKay quiver associated to the coaction.

Sl(2,C)×D symmetry and conformal primary basis for massless fields

Alternative to the embedding formalism, we provide a group theoretic approach to the conformal primary basis for the massless field with arbitrary helicity. To this end, we first point out that sl(2,C) isometry gets enhanced to sl(2,C)×D symmetry for the solution space of the massless field with D the bulk dilatation. Then, associated with sl(2,C)×D symmetry, we introduce the novel quadratic Casimirs and relevant tensor/spinor fields to derive two explicit constraints on the bulk dilatation and sl(2,C) Casimirs. With this, we further argue that the candidate conformal primary basis can be constructed out of the infinite tower of the descendants of the left and right highest (lowest) conformal primary wave function of sl(2,C) Lie algebra, and the corresponding celestial conformal weights are determined by the bulk scaling dimension through solving out the exact on-shell conformal primary wave functions, where on top of the two kinds of familiar-looking on-shell conformal primary wave functions, we also obtain another set of independent on-shell conformal primary wave functions for the massless field with helicity |s|≥1. In passing, we also develop the relationship between the four-dimensional Lorentz Lie algebra and two-dimensional conformal Lie algebra from scratch, and present an explicit derivation for the two important properties associated with the conformal primary wave functions. Published by the American Physical Society 2024

Integrable Couplings and Two-Dimensional Unital Algebras

The paper aims to demonstrate that a linear expansion in a unital two-dimensional algebra can generate integrable couplings, proposing a novel approach for their construction. The integrable couplings presented encompass a range of perturbation equations and nonlinear integrable couplings. Their corresponding Lax pairs and hereditary recursion operators are explicitly detailed. Concrete applications to the KdV equation and the AKNS system of nonlinear Schrödinger equations are extensively explored.

Malcev-like binary Lie algebras of dimension 5

We study anti-commutative algebras, which are extensions of a one-dimensional algebra by a four-dimensional nilpotent Lie algebra and at the same time extensions of the two-dimensional non-Abelian Lie algebra by an abelian algebra. These algebras, just like the five-dimensional solvable Malcev algebras, have a flag of subalgebras and can therefore be considered their closest relatives. We characterize binary Lie algebras in this class, called Malcev-like algebras, playing interesting role in non-associative Lie theory. We find normal forms of their multiplications, and determine their isomorphism classes.

Invariant analysis of the two-cell tumor growth model in the brain

In this study, we explore the invariance properties of a tumor growth model involving two distinct cell populations. These populations are characterized by different diffusion coefficients but share a common killing rate. This particular model serves as a representation of tumor growth within the brain. By employing the Lie group method, we unveil a two-dimensional symmetry algebra for cases where both diffusion coefficients are allowed to vary arbitrarily. Interestingly, this method reveals a nine-dimensional symmetry algebra when the diffusion coefficients are held constant. In both scenarios involving varying and constant diffusion coefficients, we conduct similarity reductions to deduce group invariant solutions, thus elucidating the model’s behavior. Notably, our findings demonstrate that the tumor’s growth remains exponential irrespective of the presence or absence of a killing rate. This remarkable phenomenon holds for various configurations of diffusion coefficients. To validate our observations, we employ Mathematica simulations, which corroborate the model’s exponential growth behavior and emphasize the role of killing rates, diffusion coefficients, and growth rate parameters in driving this exponential trend. Also, the conserved flows and conserved quantities of the model are demonstrated.

Twisted holography without conformal symmetry

We discuss the notion of translation-invariant vacua for 2d chiral algebras and relate it to the notion of the associated variety. The two-dimensional chiral algebra associated to four-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 U(N) SYM has a conjectural holographic dual involving the B-model topological string theory. We study the effect of non-zero vacuum expectation values on the chiral algebra correlation functions and derive a holographic dual Calabi-Yau geometry. We test our proposal by a large N analysis of correlation functions of determinant operators, whose saddles can be matched with semi-classical configurations of “Giant Graviton” D-branes in the bulk.

Peculiarities of beta functions in sigma models

In this paper we consider perturbation theory in generic two-dimensional sigma models in the so-called first-order formalism, using the coordinate regularization approach. Our goal is to analyze the first-order formalism in application to β functions and compare its results with the standard geometric calculations. Already in the second loop, we observe deviations from the geometric results that cannot be explained by the regularization/renormalization scheme choices. Moreover, in certain cases the first-order calculations produce results that are not symmetric under the classical diffeomorphisms of the target space. Although we could not present the full solution to this remarkable phenomenon, we found some indirect arguments indicating that an anomaly similar to that established in supersymmetric Yang-Mills theory might manifest itself starting from the second loop. We discuss why the difference between two answers might be an infrared effect, similar to that in β functions in supersymmetric Yang-Mills theories.In addition to the generic Kähler target spaces we discuss in detail the so-called Lie-algebraic sigma models. In particular, this is the case when the perturbed field Gij¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {G}^{i\\overline{j}} $$\\end{document} is a product of the holomorphic and antiholomorphic currents satisfying two-dimensional current algebra.

σ-monogenic functions in commutative algebras

In finite-dimensional commutative associative algebra, the concept of σ-monogenic function is introduced. Necessary and sufficient conditions for σ-monogeneity have been established. In some low-dimensional algebras, with a special choice of σ, the representation of σ-monogenic functions is obtained using holomorphic functions of a complex variable. We proposed the application of σ-monogenic functions with values in two-dimensional biharmonic algebra to representation of solutions of two-dimensional biharmonic equation.

Penggunaan Software Geogebra Classic Pada Mata Pelajaran Matematika (Program Linear)

Geogebra classic as it is known today can be run through various platforms. At the beginning of the introduction and development of GeoGebra, I learned about the Makassar City Mathematics MGMP training being held by the Directorate General of the Republic of Indonesia in South Sulawesi in 2021. At that time, Geogebra Classic was only introduced as a free mathematics software that could be run via the Windows operating system. In its development, GeoGebra Classic has undergone many changes, especially now that it can be used on various platforms. At the beginning of its development, GeoGebra Classic was only used for two-dimensional geometry and algebra material. But now, GeoGebra Classic can be used for three-dimensional geometry with other capabilities. Geogebra classic that is used today is geogebra classic. Geogebra classic was originally known as geogebra 5, then geogebra 6. At the time this article was written, it was known as geogebra classic. At present, the compression of computer programs in learning mathematics is a necessity and is a mandatory requirement for students at both the elementary, secondary, and tertiary levels today. There are lots of computer programs and Android phones that can be used in learning mathematics, including the GeoGebra application. Geogebra classic is a dynamic program with a variety of facilities that are very useful for students, especially on the subject of linear programming, and can be used as a medium for learning mathematics. In addition, Geogebra classic can be used to demonstrate or visualize mathematical concepts as well as tools to construct these concepts. With this program, abstract mathematical objects can be visualized and manipulated quickly and efficiently.

Two-dimensional perfect evolution algebras over domains

We will study evolution algebras A that are free modules of dimension two over domains. We start by making some general considerations about algebras over domains: They are sandwiched between a certain essential D-submodule and its scalar extension over the field of fractions of the domain. We introduce the notion of quasiperfect algebras and we characterize the perfect and quasiperfect evolution algebras in terms of the determinant of its structure matrix. We classify the two-dimensional perfect evolution algebras over domains parametrizing the isomorphism classes by a convenient moduli set.

Two-dimensional Jordan algebras: Their classification and polynomial identities

Let F be an arbitrary field of characteristic different from 2. We classify the two-dimensional power-associative commutative algebras over F. As consequence, we obtain a classification of two-dimensional Jordan algebras over F and prove that there exists, up to isomorphism, a unique two-dimensional nonassociative Jordan algebra. The construction of this algebra can be generalized naturally to produce a Jordan algebra D with an arbitrary dimension. If F is infinite, we determine a finite basis for the polynomial identities of D, as well of all associative Jordan algebras of dimension two. We also determine the codimension sequence of all these algebras, and if the field is of characteristic zero we determine their cocharacter sequence. Hence we conclude that the n-th codimension sequence of a two-dimensional Jordan algebra is bounded by n.

二维预约当代数的Rota-Baxter算子

二维预约当代数的Rota-Baxter算子

Affine automorphisms of the universal multiplicative enveloping algebra of the two-dimensional left-symmetric algebra with zero multiplication

We construct the basis of the universal multiplicative enveloping algebra of the left-symmetric algebra. We also describe the affine automorphisms of the universal multiplicative enveloping algebra of the twodimensional left-symmetric algebra with zero multiplication.

ON PRE-HILBERT ALGEBRAS CONTAINING A NONZERO CENTRAL IDEMPOTENT SUCH THAT ‖FA‖=‖A‖ and ‖A^2‖≤‖A‖^2

This paper aims to prove that every two-dimensional real absolute valued algebra is isomorphic to C〖,C〗^*,*C, or C┴*, and if the algebra A is power commutative, then A is the third power associative. Let A be real pre-Hilbert algebra without divisors of zero; we prove that if A has dimension two and satisfying,‖a^2 ‖=‖a‖^2 for all a ∈ A then A is isomorphic to new classes of two-dimensional pre-Hilbert algebras. We also characterize the pre-Hilbert algebraic algebras without divisors of zero and containing a nonzero central idempotent f such that ‖fa‖=‖a‖ and ‖a^2 ‖≤‖a‖^2, to be flexible algebras. Furthermore, we prove that if A contains a nonzero central idempotent f such that ‖fa‖=‖a‖ and ‖a^2 ‖≤‖a‖^2 for all a in A, then these statements are equivalent: A is power commutative, A is the third power associative, A is algebraic of degree two.

Classification of a family of 4-dimensional anti-commutative algebras and their automorphisms

We determine normal forms of the multiplication of four-dimensional anti-commutative algebras over a field K of characteristic zero having an analogous family of flags of subalgebras as the four-dimensional non-Lie binary Lie algebras, and hence can be considered as the closest relatives of binary Lie algebras. These algebras are extensions of K by the 3-dimensional nilpotent Lie algebra and at the same time extensions of a two-dimensional Lie algebra by a two-dimensional abelian algebra. We describe their groups of automorphisms as extensions of a subgroup of the group of automorphisms of the three-dimensional nilpotent Lie algebra by K.

Boosting to BMS

Bondi-Metzner-Sachs (BMS) symmetries, or equivalently Conformal Carroll symmetries, are intrinsically associated to null manifolds and in two dimensions can be obtained as an Inönü-Wigner contraction of the two-dimensional (2d) relativistic conformal algebra. Instead of performing contractions, we demonstrate in this paper how this transmutation of symmetries can be achieved by infinite boosts or degenerate linear transformations on coordinates. Taking explicit cues from the worldsheet theory of null strings, we show boosting the system is equivalent to adding a current-current deformation term to the Hamiltonian. As the strength of this deformation term reaches a critical value, the classical symmetry algebra “flows” from two copies of Virasoro to the BMS algebra. We further explore the situation where the CFT coordinates are asymmetrically transformed, and degenerate limits lead to chiral theories.

Canonical Form Algorithm and Separating Systems for Two-Dimensional Algebras

For a field F, where any polynomial of degree two and three possess a root, we give an algorithm transforming any two-dimensional algebra over F to a "canonical form" in terms of structure constants of the algebra. A system of invariant conditions in form of equalities and non-equalities (inequalities) separating non-isomorphic two-dimensional algebras is given.

Superconformal Algebras and Holomorphic Field Theories

We show that four-dimensional superconformal algebras admit an infinite-dimensional derived enhancement after performing a holomorphic twist. The type of higher symmetry algebras we find is closely related to algebras studied by Faonte–Hennion–Kapranov, Hennion–Kapranov, and the second author with Gwilliam in the context of holomorphic QFT. We show that these algebras are related to the two-dimensional chiral algebras extracted from four-dimensional superconformal theories by Beem and collaborators; further deforming by a superconformal element induces the Koszul resolution of a plane in mathbb {C}^2 cong {mathbb {R}}^4. The central charges at the level of chiral algebras arise from central extensions of the higher symmetry algebras.

Dynamical symmetries of homogeneous minisuperspace models

We investigate the phase space symmetries and conserved charges of homogeneous gravitational minisuperspaces. These ($0+1$)-dimensional reductions of general relativity are defined by spacetime metrics in which the dynamical variables depend only on a time coordinate and are formulated as mechanical systems with a nontrivial field space metric (or supermetric) and effective potential. We show how to extract conserved charges for those minisuperspaces from the homothetic Killing vectors of the field space metric. In the case of two-dimensional field spaces, we exhibit a universal eight-dimensional symmetry algebra $\mathcal{A}=(\mathfrak{sl}(2,\mathbb{R})\ensuremath{\bigoplus}\mathbb{R})⨭{\mathfrak{h}}_{2}$, based on the two-dimensional Heisenberg algebra ${\mathfrak{h}}_{2}\ensuremath{\simeq}{\mathbb{R}}^{4}$. We apply this to the systematic study of the Bianchi models for homogeneous cosmology. This extends the previous results on the $\mathfrak{sl}(2,\mathbb{R})$ algebra for Friedmann-Lema\^{\i}tre-Robertson-Walker cosmology and the Poincar\'e symmetry for Kantowski-Sachs metrics describing the black hole interior. The presence of this rich symmetry structure already in minisuperspace models opens new doors toward quantization and the study of solution generating mechanisms.