Terms Of Algebra Research Articles

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.

The algebraic face of the Alexandro one point compatication

We give a characterization, in algebraic terms, of the Alexandroff one point compactication of a locally compact Hausdorff space. In fact, we prove that if (X, T) is a locally compact Hausdorff space, then (X', T') point compactication if and only if T is a maximal ideal of T'.

Homotopy Transfer and Effective Field Theory I: Tree‐level

AbstractWe use the dictionary between general field theories and strongly homotopy algebras to provide an algebraic formulation of the procedure of integrating out of degrees of freedom in terms of homotopy transfer. This includes more general effective theories in which some massive modes are kept while other modes of a comparable mass scale are integrated out, as first explored by Sen in the context of closed string field theory. We treat ‐algebras both in terms of a nilpotent coderivation and, on the dual space, in terms of a nilpotent derivation (corresponding to the BRST charge of the field theory) and provide explicit formulas for homotopy transfer. These are then shown to govern the integrating out of degrees of freedom at tree level, while the generalization to loop level will be explored in a sequel to this paper.

Equilibrium stressability of multidimensional frameworks

We prove an equilibrium stressability criterion for trivalent multidimensional frameworks. The criterion appears in different languages: (1) in terms of stress monodromies, (2) in terms of surgeries, (3) in terms of exact discrete 1-forms, and (4) in Cayley algebra terms.

Flow-equation approach to quantum systems driven by an amplitude-modulated time-periodic force

We apply the method of flow equations to describe quantum systems subject to a time-periodic drive with a time-dependent envelope. The driven Hamiltonian is expressed in terms of its constituent Fourier harmonics with amplitudes that may vary as a function of time. The time evolution of the system is described in terms of the phase-independent effective Hamiltonian and the complementary micromotion operator that are generated by deriving and solving the flow equations. These equations implement the evolution with respect to an auxiliary flow variable and facilitate a gradual transformation of the quasienergy matrix (the Kamiltonian) into a block diagonal form in the extended space. We construct a flow generator that prevents the appearance of additional Fourier harmonics during the flow, thus enabling implementation of the flow in a computer algebra system. Automated generation of otherwise cumbersome high-frequency expansions (for both the effective Hamiltonian and the micromotion) to an arbitrary order thus becomes straightforward for driven Hamiltonians expressible in terms of a finite algebra of Hermitian operators. We give several specific examples and discuss the possibility to extend the treatment to cover rapid modulation of the envelope.

Derived equivalences between skew-gentle algebras using orbifolds

Skew-gentle algebras are skew-group algebras of gentle algebras equipped with a certain $\Z_2$-action. Building on the bijective correspondence between gentle algebras and dissected surfaces, we obtain in this paper a bijection between skew-gentle algebras and certain dissected orbifolds that admit a double cover. We prove the compatibility of the $\Z_2$-action on the double cover with the skew-group algebra construction. This allows us to investigate the derived equivalence relation between skew-gentle algebras in geometric terms: We associate to each skew-gentle algebra a line field on the orbifold, and on its double cover, and interpret different kinds of derived equivalences of skew-gentle algebras in terms of diffeomorphisms respecting the homotopy class of the line fields associated to the algebras.

On C^*-norms on {{mathbb {Z}}}_2-graded tensor products

We systematically investigate C^*-norms on the algebraic graded product of {{mathbb {Z}}}_2-graded C^*-algebras. This requires to single out the notion of a compatible norm, that is a norm with respect to which the product grading is bounded. We then focus on the spatial norm proving that it is minimal among all compatible C^*-norms. To this end, we first show that commutative {{mathbb {Z}}}_2-graded C^*-algebras enjoy a nuclearity property in the category of graded C^*-algebras. In addition, we provide a characterization of the extreme even states of a given graded C^*-algebra in terms of their restriction to its even part.

On Some Homological Properties of Hypergroup Algebras with Relation to Their Character Spaces

In this paper, we study the notion of approximate biprojectivity and left φ‐biprojectivity of some Banach algebras, where φ is a character. Indeed, we show that approximate biprojectivity of the hypergroup algebra L1(K) implies that K is compact. Moreover, we investigate left φ‐biprojectivity of certain hypergroup algebras, namely, abstract Segal algebras. As a main result, we conclude that (with some mild conditions) the abstract Segal algebra B is left φ‐biprojective if and only if K is compact, where K is a hypergroup. We also study the approximate biflatness and left φ‐biflatness of hypergroup algebras in terms of amenability of their related hypergroups.

INTERPOLATION OF TABULAR FUNCTIONS FROM ONE INDEPENDENT VARIABLE USING THE TAYLOR POLYNOMIAL

A method of local interpolation of tabular functions from one independent variable using the Taylor polynomial of the nth degree in arbitrarily located interpolation nodes has been developed. This makes it possible to calculate intermediate values of tabular functions between interpolation nodes. The conducted analysis of the latest research and publications in the field of interpolation of tabular functions showed that the main part of the research is a strict theory of interpolation, i.e. clarification of its fundamental mathematical provisions. Some features of the interpolation of tabular functions from one independent variable using the Taylor polynomial of the nth degree are considered, namely: the solution algorithm and mathematical formulation of the interpolation problem are given; its formalized notation is given, as well as the matrix notation of interpolation procedures for certain values of the argument. A scalar algorithm for solving the problem of interpolation of tabular functions from one independent variable using the Taylor polynomial of the 2nd, 3rd, and 4th degrees has been developed. The simplicity and clarity of this algorithm is one of its advantages, but the algorithm is inconvenient for software implementation. The mathematical formulation of the problem of interpolation of tabular functions in terms of matrix algebra is given. The interpolation task is reduced to performing the following actions: based on the values of nodal points known from the table, it is necessary to calculate the Taylor matrix of the nth degree; based on the function values specified in the table a column vector of interpolation nodes should be formed; solve a linear system of algebraic equations, the root of which is the numerical coefficients of the Taylor polynomial of the nth degree. A method of calculating the coefficients of the interpolant, given by the Taylor polynomial of the nth degree for one independent variable has been developed. The essence of the method reduces to the product of the matrix, inverse of the Taylor matrix, which is determined by the nodal points of the tabular function, by a column vector containing the values of the interpolation nodes. Specific examples demonstrate the peculiarities of calculating the interpolant coefficients of the 2nd, 3rd and 4th degrees for one independent variable, and for each of them the interpolated value of the function at a given point is calculated. Calculations were performed in the Excel environment, which by analogy can be successfully implemented in any other computing environment.

Lie Group Approach to Dynamic-Model-Aided Navigation of Multirotor Unmanned Aerial Vehicles

In this study, the Lie group approach was used for state estimation of the dynamic-model-aided navigation (DMAN) of a small multirotor unmanned aerial vehicle. The unit quaternions constituting a Lie group called the three-sphere space (<i>S</i>&#x00B3;) were used to represent the attitude in the dynamic equations for the process and measurement models. The linearization of these models is presented in terms of Lie algebra corresponding to <i>S</i>&#x00B3;. The use of Lie algebra to describe the attitude increment conforms to the linearity assumption, on which the measurement update of the extended Kalman filter (EKF) is based. In this study, it was experimentally validated that the Lie group approach combined with DMAN performs better than the EKF that uses the conventional linearization of the process and measurement models under the assumption that the nonlinearity effect is negligible for a small attitude increment. It was demonstrated that the navigation states estimated using the proposed model are better than or comparable to those obtained using the current methods, and the proposed method significantly improves the internal properties.

Higher-order algebraic soliton solutions of the Gerdjikov–Ivanov equation: Asymptotic analysis and emergence of rogue waves

In this paper, we derive the determinant representation of higher-order algebraic soliton solutions of the Gerdjikov–Ivanov equation by using the Darboux transformation and some limit technique. Based on the asymptotic balance between different algebraic terms, we obtain the asymptotic expressions of algebraic soliton solutions with the order 2≤N≤4. It turns out that all the asymptotic solitons have the same amplitudes, most of them are located in the parabolic curves and thus have the varying velocities with the rate O(|t|−12) (except that one pair of asymptotic solitons are located in the straight lines for the odd-order cases), and they exhibit the elastic interactions of the attractive type. In addition, we reveal that the transient rogue waves are generated in the soliton-interaction region and the peak value is exactly N times the amplitude of individual soliton.

Vector Symbolic Architectures for Context-Free Grammars

Vector symbolic architectures (VSA) are a viable approach for the hyperdimensional representation of symbolic data, such as documents, syntactic structures, or semantic frames. We present a rigorous mathematical framework for the representation of phrase structure trees and parse trees of context-free grammars (CFG) in Fock space, i.e. infinite-dimensional Hilbert space as being used in quantum field theory. We define a novel normal form for CFG by means of term algebras. Using a recently developed software toolbox, called FockBox, we construct Fock space representations for the trees built up by a CFG left-corner (LC) parser. We prove a universal representation theorem for CFG term algebras in Fock space and illustrate our findings through a low-dimensional principal component projection of the LC parser state. Our approach could leverage the development of VSA for explainable artificial intelligence (XAI) by means of hyperdimensional deep neural computation.

Linking the special orthogonal algebra [formula omitted] and the tetrahedron algebra ⊠

In 2007, B. Hartwig and Terwilliger found a presentation for the three-point sl2 loop algebra in terms of generators and relations. To obtain this presentation, they defined a Lie algebra ⊠ by generators and relations and established an isomorphism from ⊠ to three-point sl2 loop algebra. Essentially, ⊠ has six generators which can be naturally identified with the six edges of the tetrahedron. In fact, each face of the tetrahedron has three surrounding edges which generate a subalgebra of ⊠ that is isomorphic to sl2. It is interesting to know whether a direct sum of finitely many copies of sl2 (e.g., special orthogonal algebra so4) captures the bracket relations of the generators of ⊠. Here, we show that there exists a Lie algebra homomorphism ϕ:⊠→so4 which can be extended to a homomorphism ϕ:⊠→L where L is a direct sum of finitely many copies of sl2. We construct a finite-dimensional so4-module which is viewed as a ⊠-module via the homomorphism ϕ. We show how this so4-module is related to Krawtchouk polynomials. This paper is inspired by and is an extension of the work of Nomura and Terwilliger (2012) [19].

PBW parametrizations and generalized preprojective algebras

Geiß-Leclerc-Schröer (2017) [24] has introduced a notion of generalized preprojective algebras associated with generalized Cartan matrices and their symmetrizers. These algebras realize crystal structures on the set of maximal dimensional irreducible components of the nilpotent varieties (Geiss et al. 2018, [26]). For general finite types, we give stratifications of these components via partial orders of torsion classes in module categories of generalized preprojective algebras in terms of Weyl groups. In addition, we realize Mirković-Vilonen polytopes from generic modules of these components, and give an identification as crystals between the set of Mirković-Vilonen polytopes and the set of maximal dimensional irreducible components. This generalizes results of Baumann-Kamnitzer (2012) [8] and Baumann-Kamnitzer-Tingley (2014) [10].

Two-step nilpotent Leibniz algebras

In this paper we give a complete classification of two-step nilpotent Leibniz algebras in terms of Kronecker modules associated with pairs of bilinear forms. Among these, we describe the complex and the real case of the indecomposable Heisenberg Leibniz algebras as a generalization of the classical (2n+1)-dimensional Heisenberg Lie algebra h2n+1. Then we use the Leibniz algebras - Lie local racks correspondence proposed by S. Covez to show that nilpotent real Leibniz algebras have always a global integration. As an application, we integrate the indecomposable nilpotent real Leibniz algebras with one-dimensional commutator ideal. Finally we show that every Lie quandle integrating a Leibniz algebra is induced by the conjugation of a Lie group and the Leibniz algebra is the Lie algebra of that Lie group.

Model checking strategy-controlled systems in rewriting logic

Rewriting logic and its implementation Maude are an expressive framework for the formal specification and verification of software and other kinds of systems. Concurrency is naturally represented by nondeterministic local transformations produced by the application of rewriting rules over algebraic terms in an equational theory. Some aspects of the global behavior of the systems or additional constraints sometimes require restricting this nondeterminism. Rewriting strategies are used as a higher-level and modular resource to cleanly capture these requirements, which can be easily expressed in Maude with an integrated strategy language. However, strategy-aware specifications cannot be verified with the builtin LTL model checker, making strategies less useful and attractive. In this paper, we discuss model checking for strategy-controlled systems, and present a strategy-aware extension of the Maude LTL model checker. The expressivity of the strategy language is discussed in relation to model checking, the model checker is illustrated with multiple application examples, and its performance is compared.

Automatization of Ternary Boolean Algebras

Summary The main aim of this article is to introduce formally ternary Boolean algebras (TBAs) in terms of an abstract ternary operation, and to show their connection with the ordinary notion of a Boolean algebra, already present in the Mizar Mathematical Library [2]. Essentially, the core of this Mizar [1] formalization is based on the paper of A.A. Grau “Ternary Boolean Algebras” [7]. The main result is the single axiom for this class of lattices [12]. This is the continuation of the articles devoted to various equivalent axiomatizations of Boolean algebras: following Huntington [8] in terms of the binary sum and the complementation useful in the formalization of the Robbins problem [5], in terms of Sheffer stroke [9]. The classical definition ([6], [3]) can be found in [15] and its formalization is described in [4]. Similarly as in the case of recent formalizations of WA-lattices [14] and quasilattices [10], some of the results were proven in the Mizar system with the help of Prover9 [13], [11] proof assistant, so proofs are quite lengthy. They can be subject for subsequent revisions to make them more compact.

Staggered Semi-Implicit Hybrid Finite Volume/Finite Element Schemes for Turbulent and Non-Newtonian Flows

This paper presents a new family of semi-implicit hybrid finite volume/finite element schemes on edge-based staggered meshes for the numerical solution of the incompressible Reynolds-Averaged Navier–Stokes (RANS) equations in combination with the k−ε turbulence model. The rheology for calculating the laminar viscosity coefficient under consideration in this work is the one of a non-Newtonian Herschel–Bulkley (power-law) fluid with yield stress, which includes the Bingham fluid and classical Newtonian fluids as special cases. For the spatial discretization, we use edge-based staggered unstructured simplex meshes, as well as staggered non-uniform Cartesian grids. In order to get a simple and computationally efficient algorithm, we apply an operator splitting technique, where the hyperbolic convective terms of the RANS equations are discretized explicitly at the aid of a Godunov-type finite volume scheme, while the viscous parabolic terms, the elliptic pressure terms and the stiff algebraic source terms of the k−ε model are discretized implicitly. For the discretization of the elliptic pressure Poisson equation, we use classical conforming P1 and Q1 finite elements on triangles and rectangles, respectively. The implicit discretization of the viscous terms is mandatory for non-Newtonian fluids, since the apparent viscosity can tend to infinity for fluids with yield stress and certain power-law fluids. It is carried out with P1 finite elements on triangular simplex meshes and with finite volumes on rectangles. For Cartesian grids and more general orthogonal unstructured meshes, we can prove that our new scheme can preserve the positivity of k and ε. This is achieved via a special implicit discretization of the stiff algebraic relaxation source terms, using a suitable combination of the discrete evolution equations for the logarithms of k and ε. The method is applied to some classical academic benchmark problems for non-Newtonian and turbulent flows in two space dimensions, comparing the obtained numerical results with available exact or numerical reference solutions. In all cases, an excellent agreement is observed.

The algebra of 2 × 2 upper triangular matrices as a commutative algebra: Gradings, graded polynomial identities and Specht property

Let UJ2 be the Jordan algebra of 2×2 upper triangular matrices. This paper is devoted to continue the description given by recent works about the gradings and graded polynomial identities on UJ2(K) when K is an infinite field of characteristic 2. Due to the definition of Jordan algebras in terms of the commutative and Jordan identities being unsuitable in characteristic 2, we decided to study the gradings of the non-associative commutative algebra of 2×2 upper triangular matrices UT2=(UT2(K),∘) with the product x∘y=xy+yx. More precisely, fixed K a field of characteristic 2, we classify the gradings of (UT2(K),∘) and also, given an arbitrary grading, we calculate the generators of the ideals of graded identities and give a positive answer to the Specht property for the variety of commutative algebras generated by (UT2(K),∘) in each grading when K is infinite.

Fibred-categorical obstruction theory

We set up a fibred categorical theory of obstruction and classification of morphisms that specialises to the one of monoidal functors between categorical groups and also to the Schreier-Mac Lane theory of group extensions. Further applications are provided to crossed extensions and crossed bimodule butterflies, with in particular a classification of non-abelian extensions of unital associative algebras in terms of Hochschild cohomology.