Algebraic Objects Research Articles

A motivic global Torelli theorem for isogenous K3 surfaces

We prove that the Chow motives of twisted derived equivalent K3 surfaces are isomorphic, not only as Chow motives (due to Huybrechts), but also as Frobenius algebra objects. Combined with a recent result of Huybrechts, we conclude that two complex projective K3 surfaces are isogenous (i.e. their second rational cohomology groups are Hodge isometric) if and only if their Chow motives are isomorphic as Frobenius algebra objects; this can be regarded as a motivic Torelli-type theorem. We ask whether, more generally, twisted derived equivalent hyper-Kähler varieties have isomorphic Chow motives as (Frobenius) algebra objects and in particular isomorphic graded rational cohomology algebras. In the appendix, we justify introducing the notion of “Frobenius algebra object” by showing the existence of an infinite family of K3 surfaces whose Chow motives are pairwise non-isomorphic as Frobenius algebra objects but isomorphic as algebra objects. In particular, K3 surfaces in that family are pairwise non-isogenous but have isomorphic rational Hodge algebras.

Uprolling unrolled quantum groups

We construct families of commutative (super) algebra objects in the category of weight modules for the unrolled restricted quantum group [Formula: see text] of a simple Lie algebra [Formula: see text] at roots of unity, and study their categories of local modules. We determine their simple modules and derive conditions for these categories being finite, non-degenerate, and ribbon. Motivated by numerous examples in the [Formula: see text] case, we expect some of these categories to compare nicely to categories of modules for vertex operator algebras. We focus in particular on examples expected to correspond to the higher rank triplet vertex algebra [Formula: see text] of Feigin and Tipunin and the [Formula: see text] algebras.

Synchronous linear constraint system games

Mathematical models of quantum mechanics can be studied and distinguished using nonlocal games. We discuss a class of nonlocal games called synchronous linear constraint system (syncLCS) games. We unify two algebraic approaches to studying syncLCS games and relate these games to nonlocal games played on graphs, known as graph isomorphism games. In more detail, syncLCS games are nonlocal games that verify whether or not two players share a solution to a given system of equations. Two algebraic objects associated with these games encode information about the existence of perfect strategies. They are called the game algebra and the solution group. Here, we show that these objects are essentially the same, i.e., the game algebra is a suitable quotient of the group algebra of the solution group. We also demonstrate that syncLCS games are equivalent to graph isomorphism games on a pair of graphs parameterized by the linear system.

Investigation of Representation Theory for Algebraic and Geometric Applications

Understanding the algebraic and geometric structures that form in diverse mathematical areas depends heavily on the study of representation theory. The significance and uses of representation theory in both algebra and geometry are briefly discussed in this abstract.The primary goal of representation theory is to understand how linear transformations on vector spaces can represent abstract algebraic objects like groups, rings, and algebras. Representation theory offers a strong framework to analyse and interact with these structures using the methods and tools of linear algebra by linking algebraic structures with linear transformations.The representation theory has significant effects on algebra. Through the examination of the representations that go along with a group, it allows us to examine its composition and behaviour. One can learn more about the internal structures and underlying symmetries of groups by breaking representations down into irreducible parts. This has implications for number theory, combinatorics, and quantum physics, among other fields.Understanding symmetries and transformations of geometric objects in geometry depends critically on representation theory. The investigation of shape and space symmetry is made possible by representation theory, which links geometric objects with linear transformations. This has uses in a variety of disciplines, including physics' study of symmetry groups, differential geometry, and crystallography.A crucial area of research, representation theory has extensive uses in both algebra and geometry. It is indispensable in many branches of mathematics and opens up new directions for research and discovery because it can reveal the underlying structures and symmetries of abstract algebraic objects.

Free product on semihypergroups

In a previous paper [1], we initiated a systematic study of semihypergroups and had a thorough discussion about some important analytic and algebraic objects associated to this class of objects. In this paper, we investigate free structures on the category of semihypergroups. We show that the natural free product structure along with the natural topology, although fails to give a free product for topological groups, works well on a vast non-trivial class of ‘pure’ semihypergroups containing most of the well-known examples including non-trivial coset and orbit spaces.

Recollements of derived categories I: Construction from exact contexts

Recollements were introduced originally by Beilinson, Bernstein and Deligne to study the derived categories of perverse sheaves, and nowadays become very powerful in understanding relationship among three algebraic, geometric or topological objects. The purpose of this series of papers is to study recollements in terms of derived module categories and homological ring epimorphisms, and then to apply our results to both representation theory and algebraic K-theory. In this paper we present a new and systematic method to construct recollements of derived module categories. For this aim, we introduce a new ring structure, called the noncommutative tensor product, and give necessary and sufficient conditions for noncommutative localizations which appears often in representation theory, topology and K-theory, to be homological. The input of our machinery is an exact context which can be easily obtained from a rigid morphism that exists in very general circumstances. The output is a recollement of derived module categories of rings in which the noncommutative tensor product of an exact context plays a crucial role. Thus we obtain a large variety of new recollements from commutative and noncommutative localizations, ring epimorphisms and extensions.

Collectives of Automata in Finitely Generated Groups

The present paper is devoted to traversing a maze by a collective of automata. This part of automata theory gave rise to a fairly wide range of diverse problems ([1], [2]), including those related to problems of the theory of computational complexity and probability theory. It turns out that the consideration of complicated algebraic objects, such as Burnside groups, can be interesting in this context. In the paper, we show that the Cayley graph a finitely generated group cannot be traversed by a collective of automata if and only if the group is infinite and its every element is periodic.

Right Angled Artin Groups and partial commutation, old and new

We systematically treat algebraic objects with free partially commuting generators and give short and modern proofs of the various relations between them. These objects include right angled Artin groups, polynomial rings, Lie algebras, and restricted Lie algebras in partially commuting free generators. In particular, we compute the p -central and exponent- p series of all right angled Artin groups, and compute the dimensions of their subquotients. We also describe their associated Lie algebras, and relate them to the cohomology ring of the group as well as to polynomial and power series rings in partially commuting variables. We finally show how the growth series of these various objects are related to each other.

A $p$-adically entire function with integral values on $\mathbb {Q}_p$ and entire liftings of the $p$-divisible group $\mathbb {Q}_p/\mathbb {Z}_p$

We give a self-contained proof of the fact that, for any prime number $p$, there exists a power series $$\Psi= \Psi_p(T) \in T + T^2\Z[[T]] $$ which trivializes the addition law of the formal group of Witt covectors is $p$-adically entire and assumes values in $\Z_p$ all over $\Q_p$. We actually generalize, following a suggestion of M. Candilera, the previous facts to any fixed unramified extension $\Q_q$ of $\Q_p$ of degree $f$, where $q = p^f$. We show that $\Psi = \Psi_q$ provides a quasi-finite covering of the Berkovich affine line $\A^1_{\Q_p}$ by itself. We prove in section 3 new strong estimates for the growth of $\Psi$, in view of the application to $p$-adic Fourier expansions on $\Q_p$. We locate the zeros of $\Psi$ and to obtain its product expansion. We reconcile the present discussion (for $q =p$) with a previous formal group proof which takes place in the Frechet algebra $\Q_p\{x\}$ of the analytic additive group $\G_{a,\Q_p}$ over $\Q_p$. We show that, for any $\lambda \in \Q_p^\times$, the closure $\sE_\lambda^\circ$ of $\Z_p[\Psi(p^ix/\lambda)\,|\,i=0,1,\dots]$ in $\Q_p\{x\}$ is a Hopf algebra object in the category of Frechet $\Z_p$-algebras. The special fiber of $\sE_\lambda^\circ$ is the affine algebra of the $p$-divisible group $\Q_p/p \lambda \Z_p$ over $\F_p$, while $\sE_\lambda^\circ [1/p]$ is dense in $\Q_p\{x\}$. From $\Z_p[\Psi(\lambda x)\,|\,\lambda \in \Q_p^\times]$ we construct a $p$-adic analog $\AP_{\Q_p}(\Sigma_\rho)$ of the algebra of Dirichlet series holomorphic in a strip $(-\rho, \rho) \times i \R \subset \C$. We start developing this analogy. It turns out that the Banach algebra of almost periodic functions on $\Q_p$ identifies with the topological ring of germs of holomorphic almost periodic functions on strips around $\Q_p$.

Direct Tracking of the Pareto Front of a Multi-Objective Optimization Problem

In this paper, some methodologies aimed at the identification of the Pareto front of a multi-objective optimization problem are presented and applied. Three different approaches are presented: local sampling, Pareto front resampling and Normal Boundary Intersection (NBI). A first approximation of the Pareto front is obtained by a regular sampling of the design space, and then the Pareto front is improved and enriched using the other two above mentioned techniques. A detailed Pareto front is obtained for an optimization problem where algebraic objective functions are applied, also in comparison with standard techniques. Encouraging results are also obtained for two different ship design problems. The use of the algebraic functions allows for a comparison with the real Pareto front, correctly detected. The variety of the ship design problems allows for a generalization of the applicability of the methodology.

PSEUDOCHARACTERS OF HOMOMORPHISMS INTO CLASSICAL GROUPS

A $GL_d$-pseudocharacter is a function from a group $\Gamma$ to a ring $k$ satisfying polynomial relations which make it "look like" the character of a representation. When $k$ is an algebraically closed field, Taylor proved that $GL_d$-pseudocharacters of $\Gamma$ are the same as degree-$d$ characters of $\Gamma$ with values in $k$, hence are in bijection with equivalence classes of semisimple representations $\Gamma \rightarrow GL_d(k)$. Recently, V. Lafforgue generalized this result by showing that, for any connected reductive group $H$ over an algebraically closed field $k$ of characteristic 0 and for any group $\Gamma$, there exists an infinite collection of functions and relations which are naturally in bijection with $H^0(k)$-conjugacy classes of semisimple representations $\Gamma \rightarrow H(k)$. In this paper, we reformulate Lafforgue's result in terms of a new algebraic object called an FFG-algebra. We then define generating sets and generating relations for these objects and show that, for all $H$ as above, the corresponding FFG-algebra is finitely presented. Hence we can always define $H$-pseudocharacters consisting of finitely many functions satisfying finitely many relations. Next, we use invariant theory to give explicit finite presentations of the FFG-algebras for (general) orthogonal groups, (general) symplectic groups, and special orthogonal groups. Finally, we use our pseudocharacters to answer questions about conjugacy vs. element-conjugacy of representations, following Larsen.

A note on a method for constructing uniform topologies on algebras associated with logical systems

In the recent years, many authors have used a single method for equipping algebraic structures with uniformities which are induced by families of algebraic objects. This paper is devoted to a description of this well-known method in general, and prov

English

We propose a motivic generalization of rational homotopy types. The algebraic invariants we study are defined as algebra objects in the category of mixed motives. This invariant plays a role of Sullivan's polynomial de Rham algebras. Another main notion is that of cotangent motives. Our main objective is to investigate the topological realization of these invariants and study their structures. Applying these machineries and the Tannakian theory, we construct actions of a derived motivic Galois group on rational homotopy types. Thanks to this, we deduce actions of the motivic Galois group of pro-unipotent completions of homotopy groups.

Relations between Clifford Algebra and Dirac Matrices in the Presence of Families

The internal degrees of freedom of fermions are in the spin-charge-family theory described by the Clifford algebra objects, which are superposition of an odd number of γ a ’s. Arranged into irreducible representations of “eigenvectors” of the Cartan subalgebra of the Lorentz algebra S a b ( = i 2 γ a γ b | a ≠ b ) these objects form 2 d 2 − 1 families with 2 d 2 − 1 family members each. Family members of each family offer the description of all the observed quarks and leptons and antiquarks and antileptons, appearing in families. Families are reachable by S ˜ a b = 1 2 γ ˜ a γ ˜ b | a ≠ b . Creation operators, carrying the family member and family quantum numbers form the basis vectors. The action of the operators γ a ’s, S a b , γ ˜ a ’s and S ˜ a b , applying on the basis vectors, manifests as matrices. In this paper the basis vectors in d = ( 3 + 1 ) Clifford space are discussed, chosen in a way that the matrix representations of γ a and of S a b coincide for each family quantum number, determined by S ˜ a b , with the Dirac matrices. The appearance of charges in Clifford space is discussed by embedding d = ( 3 + 1 ) space into d = ( 5 + 1 ) -dimensional space. The achievements and predictions of the spin-charge-family theory is also shortly presented.

Unifying semantic foundations for automated verification tools in Isabelle/UTP

The growing complexity and diversity of models used in the engineering of dependable systems implies that a variety of formal methods, across differing abstractions, paradigms, and presentations, must be integrated. Such an integration relies on unified semantic foundations for the various notations, and co-ordination of a variety of automated verification tools. The contribution of this paper is Isabelle/UTP, an implementation of Hoare and He's Unifying Theories of Programming, a framework for unification of formal semantics. Isabelle/UTP permits the mechanisation of computational theories for diverse paradigms, and their use in constructing formalised semantic models. These can be further applied in the development of verification tools, harnessing Isabelle's proof automation facilities. Several layers of mathematical foundations are developed, including lenses to model variables and state spaces as algebraic objects, alphabetised predicates and relations to model programs, including algebraic and axiomatic semantics, proof tools for Hoare logic and refinement calculus, and UTP theories to encode computational paradigms.

Cocycle deformations and Galois objects of semisimple Hopf algebras of dimension 16

In this article, we determine cocycle deformations and Galois objects of non-commutative and non-cocommutative semisimple Hopf algebras of dimension 16. We show that these Hopf algebras are pairwise twist inequivalent mainly by calculating their higher Frobenius-Schur indicators, and that except three Hopf algebras which are cocycle deformations of dual group algebras, none of them admit non-trivial cocycle deformations.

Minimal linear codes arising from blocking sets

Minimal linear codes are algebraic objects which gained interest in the last 20 years, due to their link with Massey’s secret sharing schemes. In this context, Ashikhmin and Barg provided a useful and a quite easy-to-handle sufficient condition for a linear code to be minimal, which has been applied in the construction of many minimal linear codes. In this paper, we generalize some recent constructions of minimal linear codes which are not based on Ashikhmin–Barg’s condition. More combinatorial and geometric methods are involved in our proofs. In particular, we present a family of codes arising from particular blocking sets, which are well-studied combinatorial objects. In this context, we will need to define cutting blocking sets and to prove some of their relations with other notions in blocking sets’ theory. At the end of the paper, we provide one explicit family of cutting blocking sets and related minimal linear codes, showing that infinitely many of its members do not satisfy the Ashikhmin–Barg’s condition.

A preconditioned, multilevel Mimetic Finite Difference method for solving the pressure-velocity system

A Mimetic Finite Difference (MFD) implementation is proposed to solve the pressure-velocity system iteratively. The recursive procedure only solves for the velocity variable, while the pressure is calculated by direct substitution, simplifying the problem. The symmetric, positive definite system obtained is solved with an innovative implementation of a pre-conditioned, multilevel, conjugate gradient algorithm, where coarse algebraic objects are easily computed from the corresponding fine level ones, based on fundamental geometric properties of the MFD technology. A succinct eigen-analysis shows second order convergence on the velocity variable for Cartesian grids, which is corroborated numerically. Also, results consider different topologies and different medium properties, showing the applicability of the method and optimal convergence rates for the velocity on Cartesian and h2-uniform curved grids.

GLOBAL ACTIONS AND VECTOR -THEORY

Purely algebraic objects like abstract groups, coset spaces, and G-modules do not have a notion of hole as do analytical and topological objects. However, equipping an algebraic object with a global action reveals holes in it and thanks to the homotopy theory of global actions, the holes can be described and quantified much as they are in the homotopy theory of topological spaces. Part I of this article, due to the first author, starts by recalling the notion of a global action and describes in detail the global actions attached to the general linear, elementary, and Steinberg groups. With these examples in mind, we describe the elementary homotopy theory of arbitrary global actions, construct their homotopy groups, and revisit their covering theory. We then equip the set $Um_{n}(R)$ of all unimodular row vectors of length $n$ over a ring $R$ with a global action. Its homotopy groups $\unicode[STIX]{x1D70B}_{i}(Um_{n}(R)),i\geqslant 0$ are christened the vector $K$ -theory groups $K_{i+1}(Um_{n}(R)),i\geqslant 0$ of $Um_{n}(R)$ . It is known that the homotopy groups $\unicode[STIX]{x1D70B}_{i}(\text{GL}_{n}(R))$ of the general linear group $\text{GL}_{n}(R)$ viewed as a global action are the Volodin $K$ -theory groups $K_{i+1,n}(R)$ . The main result of Part I is an algebraic construction of the simply connected covering map $\mathit{StUm}_{n}(R)\rightarrow \mathit{EUm}_{n}(R)$ where $\mathit{EUm}_{n}(R)$ is the path connected component of the vector $(1,0,\ldots ,0)\in Um_{n}(R)$ . The result constructs the map as a specific quotient of the simply connected covering map $St_{n}(R)\rightarrow E_{n}(R)$ of the elementary global action $E_{n}(R)$ by the Steinberg global action $St_{n}(R)$ . As expected, $K_{2}(Um_{n}(R))$ is identified with $\text{Ker}(\mathit{StUm}_{n}(R)\rightarrow \mathit{EUm}_{n}(R))$ . Part II of the paper provides an exact sequence relating stability for the Volodin $K$ -theory groups $K_{1,n}(R)$ and $K_{2,n}(R)$ to vector $K$ -theory groups.

An Interval Arithmetic Approach for Multilevel Converter Harmonic Minimization Using Parseval’s Theorem

Multilevel converters are used for DC/AC power supply conversion, which is often applied in electric vehicle (EV) motor drives. AC conversion is done by a stepped output voltage, which provides a near-sinusoidal voltage with its fundamental frequency, but contains some higher harmonics. The elimination of several harmonics is fully implemented and well described in numerous publications, see Chiasson et al. (2003, 2004, 2005); Li et al. (2010); Tarisciotti et al. (2014), and Majed et al. (2014). In these papers the first set of undesired harmonics was eliminated, which in general was done by solving an equivalent system of equations using different methods such as resultants, Newton-Raphson (Chiasson et al. (2003)) and Optimal Minimization of Total Harmonic Distortion (OMTHD) technique, see Li et al. (2010). Higher harmonics stayed unrecognized to these optimization algorithms and delivered an undesired power spectrum to the total harmonic distortion (THD) of AC conversion.This paper presents a novel approach to the global THD-optimization of three-phase systems taking into account all harmonics up to infinity. This global optimization is implemented using interval arithmetic, see Hansen and Walster (2003), which neither need a convex objective function nor continuous-differentiable function. Interval arithmetic computes guaranteed intervals containing the global minima. The optimum is computed with an algebraic objective function, which is derived from Parseval’s theorem on a 2π periodic function.