General Algebra Research Articles

The spectrum properties of an integrable G2 invariant vertex model

This paper is concerned with the study of properties of the exact solution of the fundamental integrable G2 vertex model. The model R-matrix and respective spin chain are presented in terms of the basis generators of the G2 Lie algebra. This formulation permits us to relate the number of the Bethe roots of the respective Bethe equations with the eigenvalues of the U(1) conserved charges from the Cartan subalgebra of G2. The Bethe equations are solved by a peculiar string structure which combines complex three-strings with real roots allowing us to determine the bulk properties in the thermodynamic limit. We argue that G2 spin chain is gapless but the low-lying excitations have two different speeds of sound and the underlying continuum limit is therefore not strictly Lorentz invariant. We have investigated the finite-size corrections to the ground state energy and proposed that the critical properties of the system should be governed by the product of two c=1 conformal field theories. By combining numerical and analytical methods we have computed the bulk free-energy of the G2 vertex model. We found that there are three regimes in the spectral parameter in which the free-energy is limited and continuous. There exists however at least two sharp corner points in which the bulk-free energy is not differentiable.

Menger hypercompositional algebras represented by medial n-ary hyperoperations

The necessary and sufficient conditions under which a Menger algebra can be isomorphically represented by medial n-ary operations are proposed. Since a Menger hypercompositional algebra can be regarded as a generalization of a Menger algebra, for this reason, the situation for medial hyperoperations is further examined and a representation theorem of Menger hypercompositional algebras by such concepts is proved.

Monotonic multi-state quantum f-divergences

We use the Tomita–Takesaki modular theory and the Kubo–Ando operator mean to write down a large class of multi-state quantum f-divergences and prove that they satisfy the data processing inequality. For two states, this class includes the (α, z)-Rényi divergences, the f-divergences of Petz, and the Rényi Belavkin-Staszewski relative entropy as special cases. The method used is the interpolation theory of non-commutative Lωp spaces, and the result applies to general von Neumann algebras, including the local algebra of quantum field theory. We conjecture that these multi-state Rényi divergences have operational interpretations in terms of the optimal error probabilities in asymmetric multi-state quantum state discrimination.

Finite-Dimensional Approximations and Semigroup Coactions for Operator Algebras

Abstract The residual finite-dimensionality of a ${\textrm{C}}^{\ast }$-algebra is known to be encoded in a topological property of its space of representations, stating that finite-dimensional representations should be dense therein. We extend this paradigm to general (possibly non-self-adjoint) operator algebras. While numerous subtleties emerge in this greater generality, we exhibit novel tools for constructing finite-dimensional approximations. One such tool is a notion of a residually finite-dimensional coaction of a semigroup on an operator algebra, which allows us to construct finite-dimensional approximations for operator algebras of functions and operator algebras of semigroups. Our investigation is intimately related to the question of when residual finite-dimensionality of an operator algebra is inherited by its maximal ${\textrm{C}}^{\ast }$-cover, which we establish in many cases of interest.

Ke Li’s Lemma for Quantum Hypothesis Testing in General Von Neumann Algebras

A lemma stated by Li (Ann Statist 42:171-189, 2014) has been used in, for example, Nilanjana et al. (J Math Phys 57:062207, 2016), Kaur et al. (Phys Rev A 96:062318, 2017), Rouzé, Datta (IEEE Trans Inform Theory 64:593–612, 2018), Tomamichel, Tan (Comm Math Phys 338:103-137, 2015) and Wilde et al. (IEEE Trans Inform Theory 63:1792-1817, 2017) for various tasks in quantum hypothesis testing, data compression with quantum side information or quantum key distribution. This lemma was originally proven in finite dimension, with a direct extension to type I von Neumann algebras. Here, we show that the use of modular theory allows to give more transparent meaning to the objects constructed by the lemma, and to prove it for general von Neumann algebras. This yields a new proof of quantum Stein’s lemma with slightly weaker assumption, as well as immediate generalizations of its second-order asymptotics, for example the main results in Nilanjana et al. (J Math Phys 57:062207, 2016) and Li (Ann Statist 42:171-189, 2014).

Elliptic R-matrices and Feigin and Odesskii’s elliptic algebras

The algebras $$Q_{n,k}(E,\tau )$$ introduced by Feigin and Odesskii as generalizations of the 4-dimensional Sklyanin algebras form a family of quadratic algebras parametrized by coprime integers $$n>k\ge 1$$ , a complex elliptic curve E, and a point $$\tau \in E$$ . The main result in this paper is that $$Q_{n,k}(E,\tau )$$ has the same Hilbert series as the polynomial ring on n variables when $$\tau $$ is not a torsion point. We also show that $$Q_{n,k}(E,\tau )$$ is a Koszul algebra, hence of global dimension n when $$\tau $$ is not a torsion point, and, for all but countably many $$\tau $$ , $$Q_{n,k}(E,\tau )$$ is Artin–Schelter regular. The proofs use the fact that the space of quadratic relations defining $$Q_{n,k}(E,\tau )$$ is the image of an operator $$R_{\tau }(\tau )$$ that belongs to a family of operators $$R_{\tau }(z):{\mathbb {C}}^n\otimes {\mathbb {C}}^n\rightarrow {\mathbb {C}}^n\otimes {\mathbb {C}}^n$$ , $$z\in {\mathbb {C}}$$ , that (we will show) satisfy the quantum Yang–Baxter equation with spectral parameter.

On implicative BE algebras

We consider some generalizations of BCK algebras (RML, BE, aBE, BE** and aBE** algebras). We investigate the property of implicativity for these algebras. We prove that for any implicative BE** algebra the commutativity property is equivalent to the property of antisymmetry and show that implicative aBE** algebras are commutative BCK algebras. We also show that the class of implicative BE** algebras is a variety.

The Strong Converse Exponent of Discriminating Infinite-Dimensional Quantum States

The sandwiched Rényi divergences of two finite-dimensional density operators quantify their asymptotic distinguishability in the strong converse domain. This establishes the sandwiched Rényi divergences as the operationally relevant ones among the infinitely many quantum extensions of the classical Rényi divergences for Rényi parameter alpha >1. The known proof of this goes by showing that the sandwiched Rényi divergence coincides with the regularized measured Rényi divergence, which in turn is proved by asymptotic pinching, a fundamentally finite-dimensional technique. Thus, while the notion of the sandwiched Rényi divergences was extended recently to density operators on an infinite-dimensional Hilbert space (in fact, even for states of an arbitrary von Neumann algebra), these quantities were so far lacking an operational interpretation similar to the finite-dimensional case, and it has also been open whether they coincide with the regularized measured Rényi divergences. In this paper we fill this gap by answering both questions in the positive for density operators on an infinite-dimensional Hilbert space, using a simple finite-dimensional approximation technique. We also initiate the study of the sandwiched Rényi divergences, and the related problem of the strong converse exponent, for pairs of positive semi-definite operators that are not necessarily trace-class (this corresponds to considering weights in a general von Neumann algebra setting). This is motivated by the need to define conditional Rényi entropies in the infinite-dimensional setting, while it might also be interesting from the purely mathematical point of view of extending the concept of Rényi (and other) divergences to settings beyond the standard one of positive trace-class operators (positive normal functionals in the von Neumann algebra setting). In this spirit, we also discuss the definition and some properties of the more general family of Rényi (alpha ,z)-divergences of positive semi-definite operators on an infinite-dimensional separable Hilbert space.

On Born’s Reciprocal Relativity, Algebraic Extensions of the Yang and Quaplectic Algebra, and Noncommutative Curved Phase Spaces

After a brief introduction of Born’s reciprocal relativity theory is presented, we review the construction of the deformed quaplectic group that is given by the semi-direct product of U(1,3) with the deformed (noncommutative) Weyl–Heisenberg group corresponding to noncommutative fiber coordinates and momenta [Xa,Xb]≠0; [Pa,Pb]≠0. This construction leads to more general algebras given by a two-parameter family of deformations of the quaplectic algebra, and to further algebraic extensions involving antisymmetric tensor coordinates and momenta of higher ranks [Xa1a2⋯an,Xb1b2⋯bn]≠0; [Pa1a2⋯an,Pb1b2⋯bn]≠0. We continue by examining algebraic extensions of the Yang algebra in extended noncommutative phase spaces and compare them with the above extensions of the deformed quaplectic algebra. A solution is found for the exact analytical mapping of the noncommuting xμ,pμ operator variables (associated to an 8D curved phase space) to the canonical YA,ΠA operator variables of a flat 12D phase space. We explore the geometrical implications of this mapping which provides, in the classical limit, the embedding functions YA(x,p),ΠA(x,p) of an 8D curved phase space into a flat 12D phase space background. The latter embedding functions determine the functional forms of the base spacetime metric gμν(x,p), the fiber metric of the vertical space hab(x,p), and the nonlinear connection Naμ(x,p) associated with the 8D cotangent space of the 4D spacetime. Consequently, we find a direct link between noncommutative curved phase spaces in lower dimensions and commutative flat phase spaces in higher dimensions.

Boys' advantage in solving algebra word problems is mediated by spatial abilities and mathematics anxiety.

Adolescents' (n = 342, 169 boys) general algebra and algebra word problems performance were assessed in 9th grade as were intelligence, academic achievement, working memory, and spatial abilities in prior grades. The adolescents reported on their academic attitudes and anxiety and their teachers reported on their in-class attentive behavior in 7th to 9th grade. There were no sex differences on the general algebra measure or for mathematics achievement, but boys had an advantage on the algebra word problems measure (d = .51) and for spatial abilities (ds = .29 to .58). Boys had higher mathematics self-efficacy (d = .24 to .33), lower mathematics anxiety (ds = -.31 to -.53) and were less attentive in classrooms (ds = -.28 to -.37). A series of structural equation models revealed the sex difference for algebra word problems was mediated by spatial abilities and mathematics anxiety, controlling myriad confounds. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Crack the Code

This game teaches algebraic generalizations through differentiated play in pairs, small groups, or as a whole class and uses manipulatives to bridge numerical and algebraic thinking.

Computing generating sets of minimal size in finite algebras

We present an algorithm for calculating a minimal generating set of a finite algebra. Despite the fact that the problem is in NP, a single call to a SAT solver is impractical since the encoding is cubic. Instead, the proposed algorithm solves a series of smaller subproblems. The individual subproblems are formulated as integer linear programs (ILP) that are solved by an off-the-shelf solver. Our implementation shows that the proposed algorithm is highly efficient and is able to compute minimal generators for algebras of orders approximately 2000.In our experiments we focus on Moufang loops, a variety of loops with properties close to groups. For Moufang loops of prime power order, we are able to calculate a minimal generating set by another method, using theoretical results on the Frattini subloop and algorithms for permutation groups, of which some are reported here for the first time. This second method does not cover all cases, but in the covered cases it serves as a check of correctness of the ILP-based algorithm.

Non-degeneracy and uniqueness of solutions to general singular Toda systems on bounded domains

In this note we show non-degeneracy and uniqueness results for solutions of Toda systems associated to general simple Lie algebras with multiple singular sources on bounded domains. The argument is based on spectral properties of Cartan matrices and eigenvalue analysis of linearized Liouville-type problems. This seems to be the first result for this class of problems and it covers all the Lie algebras of any rank.

Quantum N-toroidal Algebras and Extended Quantized GIM Algebras of N-fold Affinization

We introduce the notion of quantum N-toroidal algebras as natural generalization of the quantum toroidal algebras as well as extended quantized GIM algebras of N-fold affinization. We show that the quantum N-toroidal algebras are quotients of the extended quantized GIM algebras of N-fold affinization, which generalizes a well-known result of Berman and Moody for Lie algebras.

Quantum geometry of Boolean algebras and de Morgan duality

We take a fresh look at the geometrization of logic using the recently developed tools of “quantum Riemannian geometry” applied in the digital case over the field \mathbb{F}_2=\{0,1\} , extending de Morgan duality to this context of differential forms and connections. The 1 -forms correspond to graphs and the exterior derivative of a subset amounts to the arrows that cross between the set and its complement. The line graph 0-1-2 has a non-flat but Ricci flat quantum Riemannian geometry. The previously known four quantum geometries on the triangle graph, of which one is curved, are revisited in terms of left-invariant differentials, as are the quantum geometries on the dual Hopf algebra, the group algebra of \mathbb{Z}_3 . For the square, we find a moduli of four quantum Riemannian geometries, all flat, while for an n -gon with n>4 we find a unique one, again flat. We also propose an extension of de Morgan duality to general algebras and differentials over \mathbb{F}_2 .

Noncommutative Yang model and its generalizations

Long time ago, Yang [Phys. Rev. 72, 874 (1947)] proposed a model of noncommutative spacetime that generalized the Snyder model to a curved background. In this paper, we review his proposal and the generalizations that have been suggested during the years. In particular, we discuss the most general algebras that contain as subalgebras both de Sitter and Snyder algebras, preserving Lorentz invariance, and are generated by a two-parameter deformation of the canonical Heisenberg algebra. We also define their realizations on quantum phase space, giving explicit examples, both exact and in terms of a perturbative expansion in deformation parameters.

The exterior graded Swiss-Cheese Operad

ABSTRACT In this paper we introduce a generalization of the exterior algebra of a vector space V. We study some of its properties and give explicit computations for the case . In particular we show the existence and uniqueness of a determinant-like function with the property that if there exists such that . We also introduce the notion of GSC-operads and show how fits in that context.

Maps preserving products of commuting elements in von Neumann algebras

A bijective map between specific structures in operator algebras is called a piecewise isomorphism if it preserves products of commuting elements in both directions. We shall show that any bicontinuous piecewise isomorphism between positive cones of invertible elements in von Neumann algebras or between unitary groups of von Neumann algebras can be described in terms of the following parameters: (i) Jordan *-isomorphism between given algebras (ii) one fixed central element in domain (or range) algebra (iii) a hermitian linear map from one algebra into the center of the other one. Especially, in case of factors any piecewise isomorphism between unitary groups is a Jordan *-isomorphism or Jordan *-isomorphism composed with inversion. This extends hitherto known results from von Neumann factors to general von Neumann algebras and brings new Jordan invariants of operator structures.

Bethe ansatz diagonalization of the Heun–Racah operator

The diagonalization of the Heun–Racah operator is studied with the help of the modified algebraic Bethe ansatz. This operator is the most general bilinear expression in two generators of the Racah algebra. A presentation of this algebra is given in terms of dynamical operators and allows the construction of Bethe vectors for the Heun–Racah operator. The associated Bethe equations are derived for both the homogeneous and inhomogeneous cases.

L-fuzzy algebraic substructure

This article aims to provide a method for defining L-fuzzy algebraic substructures on general algebras. Concretely, the properties of L-fuzzy sets are first reviewed, and their representations are then provided. Next, algebraic substructures are generalised as the closure systems on the power set of the algebra, and the properties of the prime and maximal elements in the above closure system are investigated. Based on these facts, L-fuzzy algebraic substructures with respect to the closure system are defined and studied. Two equivalence characterisations of the sup property of the ordered set L are provided using L-fuzzy substructures. Similarly, some properties of L-fuzzy prime and maximal substructures with respect to the closure system are discussed. Finally, to demonstrate the broad applicability of the theory of L-fuzzy algebraic substructures, the theory is applied to some specific algebraic structures, such as groups and pseudo MV-algebras.