Terms Of Algebra Research Articles

On the R-matrix realization of quantum loop algebras

We consider rr-matrix realization of the quantum deformations of the loop algebras \tilde{\mathfrak{g}}𝔤̃ corresponding to non-exceptional affine Lie algebras of type \widehat{\mathfrak{g}}=A^{(1)}_{N-1}𝔤̂=AN−1(1), B^{(1)}_nBn(1), C^{(1)}_nCn(1), D^{(1)}_nDn(1), A^{(2)}_{N-1}AN−1(2). For each U_q(\tilde{\mathfrak{g}})Uq(𝔤̃) we investigate the commutation relations between Gauss coordinates of the fundamental \mathbb{L}𝕃-operators using embedding of the smaller algebra into bigger one. The new realization of these algebras in terms of the currents is given. The relations between all off-diagonal Gauss coordinates and certain projections from the ordered products of the currents are presented. These relations are important in applications to the quantum integrable models.

The L∞-deformations of associative Rota–Baxter algebras and homotopy Rota–Baxter operators

A relative Rota–Baxter algebra is a triple (A, M, T) consisting of an algebra A, an A-bimodule M, and a relative Rota–Baxter operator T. Using Voronov’s derived bracket and a recent work of Lazarev, Sheng, and Tang, we construct an L∞[1]-algebra whose Maurer–Cartan elements are precisely relative Rota–Baxter algebras. By a standard twisting, we define a new L∞[1]-algebra that controls Maurer–Cartan deformations of a relative Rota–Baxter algebra (A, M, T). We introduce the cohomology of a relative Rota–Baxter algebra (A, M, T) and study infinitesimal deformations in terms of this cohomology (in low dimensions). As an application, we deduce cohomology of triangular skew-symmetric infinitesimal bialgebras and discuss their infinitesimal deformations. Finally, we define homotopy relative Rota–Baxter operators and find their relationship with homotopy dendriform algebras and homotopy pre-Lie algebras.

Free-lattice functors weakly preserve epi-pullbacks

Suppose p(x, y, z) and q(x, y, z) are terms. If there is a common “ancestor” term s(z_{1},z_{2},z_{3},z_{4}) specializing to p and q through identifying some variables p(x,y,z)≈s(x,y,z,z)q(x,y,z)≈s(x,x,y,z),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} p(x,y,z)&\\approx s(x,y,z,z)\\\\ q(x,y,z)&\\approx s(x,x,y,z), \\end{aligned}$$\\end{document}then the equation p(x,x,z)≈q(x,z,z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} p(x,x,z)\\approx q(x,z,z) \\end{aligned}$$\\end{document}is a trivial consequence. In this note we show that for lattice terms, and more generally for terms of lattice-ordered algebras, a converse is true, too. Given terms p, q, and an equation where {u_{1},ldots ,u_{m}}={v_{1},ldots ,v_{n}}, there is always an “ancestor term” s(z_{1},ldots ,z_{r}) such that p(x_{1},ldots ,x_{m}) and q(y_{1},ldots ,y_{n}) arise as substitution instances of s, whose unification results in the original equation (*). In category theoretic terms the above proposition, when restricted to lattices, has a much more concise formulation:Free-lattice functors weakly preserve pullbacksof epis. Finally, we show that weak preservation is all that we can hope for. We prove that for an arbitrary idempotent variety {{mathcal {V}}} the free-algebra functor F_{{mathcal {V}}} will not preserve pullbacks of epis unless {{mathcal {V}}} is trivial (satisfying xapprox y) or {{mathcal {V}}} contains the “variety of sets” (where all operations are implemented as projections).

Computing Riemann–Roch spaces via Puiseux expansions

Computing large Riemann–Roch spaces for plane projective curves still constitutes a major algorithmic and practical challenge. Seminal applications concern the construction of arbitrarily large algebraic geometry error correcting codes over alphabets with bounded cardinality. Nowadays such codes are increasingly involved in new areas of computer science such as cryptographic protocols and “interactive oracle proofs”. In this paper, we design a new probabilistic algorithm of Las Vegas type for computing Riemann–Roch spaces of smooth divisors, in characteristic zero, and with expected complexity exponent 2.373 (a feasible exponent for linear algebra) in terms of the input size.

On dual B-filters and dual B-subalgebras in a topological dual B-algebra

This paper introduces the notion of the \(tdB\)-algebra, presents characteristics and properties of dual \(B\)-filters and dual \(B\)-subalgebras in a \(tdB\)-algebra, and introduces the uniform topology on a dual \(B\)-algebra in terms of its dual \(B\)-subalgebras. Moreover, this paper shows that a uniform dual \(B\)-structure is a \(tdB\)-algebra.

The Minor Order of Homomorphisms via Natural Dualities

We study the minor relation for algebra homomorphims in finitely generated quasivarieties that admit a logarithmic natural duality. We characterize the minor homomorphism posets of finite algebras in terms of disjoint unions of dual partition lattices and investigate reconstruction problems for homomorphisms.

SYSTEMATIZATION OF THE FORMULAS OF THE RESONANT FERRITE ISOLATOR LOSS

Context. The problem is to systematize and improve the models of a resonance ferrite isolator in the rectangular waveguide for the antenna-feeder devices, generating, receiving, measuring microwave equipment containing ferrite decoupling devices: ferrite isolators and circulators. Objective. The goal of the work is to verify the formula for the losses of the resonant ferrite isolator in the direct and reverse directions, as well as the isolator ratio. Method. The research method of the work is a critical analysis of literary sources, which was carried out, but did not bring the desired results, since it did not allow to verify the correctness of the derivation of the formula [17]. Therefore, a number of hypotheses were put forward, what the formula might mean. The difficulty lay in the presence in the formula of the product of trigonometric functions that can be attributed to frequency properties, which was taken as an initial hypothesis, which was not subsequently confirmed. The check included transformation of formulas using mathematical physics in terms of microwave electrodynamics, trigonometry and algebra. The beginning was the formula of the classics [16], similar to the formula of [18], accepted without proof. As it is known, for the main type of wave in a rectangular waveguide, the components of the magnetic field strength, obtained as a solution to the wave equation under the boundary conditions inherent in a rectangular waveguide. One component of the magnetic field strength is along the direction of wave propagation, and the second one is in the transverse direction in the section of the waveguide are proportional to the trigonometric functions cosine and sine with the same arguments. The equality of the two components of the strengths is traditionally uses to find the plane of circular polarization where to place the ferrite isolator, and so the authors use this proportionality to trigonometric functions in their derivation, namely the formulas of trigonometric functions of a double angle, the basic trigonometric identity sine squared plus cosine squared is equal to one for replacing the propagation constants with trigonometric functions, this allows to get rid of radicals in the formulas, these radicals in the formula are due to the phenomenon of dispersion in a rectangular waveguide. The rest of the manipulations with the formula are the reduction of similar terms. Results. There was obtained analytical expressions for the losses of the resonant ferrite isolator in the forward and reverse directions, as well as the isolator ratio by strict mathematical transformations. There was performed such transformations. The ratios of the longitudinal propagation constant to the transverse propagation constant are replaced by the ratios of the trigonometric functions sine and cosine, since they are continuous as opposed to tangents and cotangents. Such a transformation allows to avoid square roots in the formula for the losses of the ferrite isolator in the forward and reverse directions, which are associated with the presence of dispersion in the waveguide, as in the formula for wavelength in the waveguide. The conversion is based on microwave electrodynamics. The formulas are used for the distribution of fields in a rectangular waveguide for the main type of wave. Further transformations consist in taking the common factor out of brackets and other arithmetic transformations. Conclusions. Тhrer was obtained results partially coincide with the well-known [17], the derivation of the formula [17] was obtained for the first time, the studies carried out allowed us to reject the hypothesis that the product of cosines and sines in the loss formula of a ferrite isolator is a frequency characteristic, it appears as a result of arithmetic transformations. To take into account the frequency range, it is used that there is circular polarization at the middle frequency, there will also be circular polarization at the extreme frequency of the range, but the plane of circular polarization will shift in comparison with the position of the plane of circular polarization at the middle frequency. That is, a peculiar system of two equations is obtained with respect to two positions of the polarization plane relative to the wide side of the rectangular waveguide section. The scientific novelty consists in systematization and generalization of the formulas of the loss of the resonance ferrite isolator, the connection between the formulas from different literature sources, both foreign and domestic, is proved, which saves time for researchers of ferrite isolators for the verification of the formula. The practical significance. It may be useful for teaching purposes and in optimization of the ferrite isolator design.

Hamiltonian Structures for Integrable Nonabelian Difference Equations

In this paper we extensively study the notion of Hamiltonian structure for nonabelian differential-difference systems, exploring the link between the different algebraic (in terms of double Poisson algebras and vertex algebras) and geometric (in terms of nonabelian Poisson bivectors) definitions. We introduce multiplicative double Poisson vertex algebras (PVAs) as the suitable noncommutative counterpart to multiplicative PVAs, used to describe Hamiltonian differential-difference equations in the commutative setting, and prove that these algebras are in one-to-one correspondence with the Poisson structures defined by difference operators, providing a sufficient condition for the fulfilment of the Jacobi identity. Moreover, we define nonabelian polyvector fields and their Schouten brackets, for both finitely generated noncommutative algebras and infinitely generated difference ones: this allows us to provide a unified characterisation of Poisson bivectors and double quasi-Poisson algebra structures. Finally, as an application we obtain some results towards the classification of local scalar Hamiltonian difference structures and construct the Hamiltonian structures for the nonabelian Kaup, Ablowitz-Ladik and Chen-Lee-Liu integrable lattices.

Lie models of homotopy automorphism monoids and classifying fibrations

Given X a finite nilpotent simplicial set, consider the classifying fibrationsX→BautG⁎(X)→BautG(X)andX→Z→Bautπ⁎(X) where G and π denote, respectively, subgroups of the free and pointed homotopy classes of free and pointed self homotopy equivalences of X which act nilpotently on H⁎(X) and π⁎(X). We give algebraic models, in terms of complete differential graded Lie algebras (cdgl's), of the rational homotopy type of these fibrations. Explicitly, if L is a cdgl model of X, there are connected sub cdgl's DerGL and DerΠL of the Lie algebra of derivations of L such that the geometrical realizations of the sequences of cdgl morphismsL→adDerGL→DerGLטsLandL→LטDerΠL→DerΠL have the rational homotopy type of the above classifying fibrations. Among the consequences we also describe in cdgl terms the Malcev Q-completion of G and π together with the rational homotopy type of the classifying spaces BG and Bπ.

A new S-box construction method meeting strict avalanche criterion

S-box is the main security component in lightweight block ciphers. Finding new algorithms and technical applications to generate S-box is still a current research topic. We proposed an S-box construction method that satisfies the strict avalanche criterion. It includes a nonlinear layer and a diffusion layer. These layers enable the constructed S-box to maintain good nonlinearity, differential uniformity, strict avalanche criterion, and a high number of algebraic terms. Secondly, we listed 20 existing classic 4 × 4 S-boxes and tested their cryptographic properties such as nonlinearity, difference uniformity, strict avalanche criterion, algebraic degree, algebraic item number, and fixed point. At the same time, the test results are compared with the proposed S-box. The results show that these cryptographic properties of our proposed S-box are superior to other S-boxes. In addition, we evaluated the proposed S-box hardware implementation overhead. To evaluate our S-box more objectively, we tested the above 20 S-boxes hardware implementation overhead under the same conditions, and the results show that the S-box we constructed achieved a good balance between cryptographic properties and hardware overhead.

On integral representations of finite groups and rings generated by character values

We study realization fields and integrality of characters of finite subgroups of [Formula: see text] and related lattices with a focus on the integrality of characters of finite groups [Formula: see text]. We are interested in the arithmetic aspects of the integral realizability of representations of finite groups, order generated by the character values, the number of minimal realization splitting fields, and in particular, consider the conditions of realizability in the terms of Hilbert symbols and quaternion algebras and some orders generated by character values over the rings of rational and algebraic integers.

An algebraic–geometric construction of ind-varieties of generalized flags

We define the class of admissible linear embeddings of flag varieties. The definition is given in the general language of algebraic geometry. We then prove that an admissible linear embedding of flag varieties has a certain explicit form in terms of linear algebra. This result enables us to show that any direct limit of admissible embeddings of flag varieties is isomorphic to an ind-variety of generalized flags as defined in Dimitrov and Penkov (IMRN No 55:2935–2953, 2004). These latter ind-varieties have been introduced in terms of the ind-group $$SL(\infty )$$ (respectively, $$O(\infty )$$ or $$Sp(\infty )$$ for isotropic generalized flags), and the current paper constructs them in purely algebraic–geometric terms.

Algebraic theory of quantum synchronization and limit cycles under dissipation

Synchronization is a phenomenon where interacting particles lock their motion and display non-trivial dynamics. Despite intense efforts studying synchronization in systems without clear classical limits, no comprehensive theory has been found. We develop such a general theory based on novel necessary and sufficient algebraic criteria for persistently oscillating eigenmodes (limit cycles) of time-independent quantum master equations. We show these eigenmodes must be quantum coherent and give an exact analytical solution for all such dynamics in terms of a dynamical symmetry algebra. Using our theory, we study both stable synchronization and metastable/transient synchronization. We use our theory to fully characterise spontaneous synchronization of autonomous systems. Moreover, we give compact algebraic criteria that may be used to prove absence of synchronization. We demonstrate synchronization in several systems relevant for various fermionic cold atom experiments.

Representations of Involutory Subalgebras of Affine Kac–Moody Algebras

We consider the subalgebras of split real, non-twisted affine Kac–Moody Lie algebras that are fixed by the Cartan–Chevalley involution. These infinite-dimensional Lie algebras are not of Kac–Moody type and admit finite-dimensional unfaithful representations. We exhibit a formulation of these algebras in terms of {mathbb {N}}-graded Lie algebras that allows the construction of a large class of representations using the techniques of induced representations. We study how these representations relate to previously established spinor representations as they arise in the theory of supergravity and work out a detailed example in the case of the affine extension of {mathfrak {e}}_8.

Semifinite Harmonic Functions on Branching Graphs

We study semifinite harmonic functions on arbitrary branching graphs. We give a detailed exposition of an algebraic method which allows one to classify semifinite indecomposable harmonic functions on some multiplicative branching graphs. It was suggested by A. Wassermann in terms of operator algebras, but we rephrase, clarify, and simplify the main arguments working only with combinatorial objects. This work was inspired by the theory of traceable factor representations of the infinite symmetric group S(∞).

Affine and formal abelian group schemes on $p$-polar rings

We show that the functor of $p$-typical co-Witt vectors on commutative algebras over a perfect field $k$ of characteristic $p$ is defined on, and in fact only depends on, a weaker structure than that of a $k$-algebra. We call this structure a $p$-polar $k$-algebra. By extension, the functors of points for any $p$-adic affine commutative group scheme and for any formal group are defined on, and only depend on, $p$-polar structures. In terms of abelian Hopf algebras, we show that a cofree cocommutative Hopf algebra can be defined on any $p$-polar $k$-algebra $P$, and it agrees with the cofree commutative Hopf algebra on a commutative $k$-algebra $A$ if $P$ is the $p$-polar algebra underlying $A$; a dual result holds for free commutative Hopf algebras on finite $k$-coalgebras.

A transform approach to polycyclic and serial codes over rings

In this paper, a transform approach is used for polycyclic and serial codes over finite local rings in the case that the defining polynomials have no multiple roots. This allows us to study them in terms of linear algebra and invariant subspaces as well as understand the duality in terms of the transform domain. We also make a characterization of when two polycyclic ambient spaces are Hamming-isometric.

The Lorentz group and the Kronecker product of matrices

The group of all complex 2 × 2 matrices with determinant one is closely related to the group of real 4 × 4 matrices representing the restricted Lorentz transformations. This relation, sometimes called the spinor map, is of fundamental importance in relativistic quantum mechanics and has applications also in general relativity. In this paper we show how the spinor map may be expressed in terms of pure matrix algebra by including the Kronecker product between matrices in the formalism. The so-obtained formula for the spinor map may be manipulated by matrix algebra and used in the study of Lorentz transformations.

Pseudo-BCK Algebras Derived from Directoids

The aim of this paper is to derive pseudo-BCK algebras from directoids and vice versa. We generalize some results proved by Ivan Chajda et al. in the case of BCK-algebras. We assign to an arbitrary pseudo-BCK algebra a semilattice-like structure and observe that this is the point where directoids are different from the semilattice-like structures. Finally, the relation between commutative deductive systems and derive directoids from a bounded pseudo-BCK(pDN) algebras and a characterization of commutative deductive systems of a bounded pseudo-BCK(pDN) algebra in terms of directoids is discussed.

Toroidal and elliptic quiver BPS algebras and beyond

The quiver Yangian, an infinite-dimensional algebra introduced recently in [1], is the algebra underlying BPS state counting problems for toric Calabi-Yau three-folds. We introduce trigonometric and elliptic analogues of quiver Yangians, which we call toroidal quiver algebras and elliptic quiver algebras, respectively. We construct the representations of the shifted toroidal and elliptic algebras in terms of the statistical model of crystal melting. We also derive the algebras and their representations from equivariant localization of three-dimensional mathcal{N} = 2 supersymmetric quiver gauge theories, and their dimensionally-reduced counterparts. The analysis of supersymmetric gauge theories suggests that there exist even richer classes of algebras associated with higher-genus Riemann surfaces and generalized cohomology theories.