Universal Algebra Research Articles

On $3$-generated axial algebras of Jordan type $\frac{1}{2}$

Axial algebras of Jordan type $\eta$ are a special type of commutative non-associative algebras. They are generated by idempotents whose adjoint operators have the minimal polynomial dividing $(x-1)x(x-\eta)$, where $\eta$ is a fixed value that is not equal to $0$ or $1$. These algebras have restrictive multiplication rules that generalize the Peirce decomposition for idempotents in Jordan algebras. A universal $3$-generated algebra of Jordan type $\frac{1}{2}$ as an algebra with $4$ parameters was constructed by I. Gorshkov and A. Staroletov. Depending on the value of the parameter, the universal algebra may contain a non-trivial form radical. In this paper, we describe all semisimple $3$-generated algebras of Jordan type $\frac{1}{2}$ over a quadratically closed field.

Universal coacting Hopf algebra of a finite dimensional Lie-Yamaguti algebra

M. E. Sweedler first constructed a universal Hopf algebra of an algebra. It is known that the dual notions to the existing ones play a dominant role in Hopf algebra theory. Yu. I. Manin and D. Tambara introduced the dual notion of Sweedler's construction in separate works. In this paper, we construct a universal algebra for a finite-dimensional Lie-Yamaguti algebra. We demonstrate that this universal algebra possesses a bialgebra structure, leading to a universal coacting Hopf algebra for a finite-dimensional Lie-Yamaguti algebra. Additionally, we develop a representation-theoretic version of our results. As an application, we characterize the automorphism group and classify all abelian group gradings of a finite-dimensional Lie-Yamaguti algebra.

Actions on classifiable C*‐algebras without equivariant property (SI)

AbstractWe exhibit examples of actions of countable discrete groups on both simple and non‐simple nuclear stably finite C*‐algebras that are tracially amenable but not amenable. We furthermore obtain that, under the additional assumption of strict comparison, amenability is equivalent to tracial amenability plus the equivariant analogue of Matui–Sato's property (SI). By virtue of this equivalence, our construction yields the first known examples of actions on classifiable C*‐algebras that do not have equivariant a over show that such actions can be chosen to absorb the trivial action on the universal UHF algebra, thus proving that equivariant ‐stability does not in general imply equivariant property (SI).

Filtered colimit elimination from Birkhoff's variety theorem

Birkhoff's variety theorem, a fundamental theorem of universal algebra, asserts that a subclass of a given algebra is definable by equations if and only if it satisfies specific closure properties. In a generalized version of this theorem, closure under filtered colimits is required. However, in some special cases, such as finite-sorted equational theories and ordered algebraic theories, the theorem holds without assuming closure under filtered colimits. We call this phenomenon “filtered colimit elimination,” and study a sufficient condition for it. We show that if a locally finitely presentable category A satisfies a noetherian-like condition, then filtered colimit elimination holds in the generalized Birkhoff's theorem for algebras relative to A.

Logic Families

AbstractA logic family is a bunch of logics that belong together in some way. First-order logic is one of the examples. Logics organized into a structure occur in abstract model theory, institution theory and in algebraic logic. Logic families play a role in adopting methods for investigating sentential logics to first-order like logics. We thoroughly discuss the notion of logic families as defined in the recent Universal Algebraic Logic book.

When left and right disagree: entropy and von Neumann algebras in quantum gravity with general AlAdS boundary conditions

Euclidean path integrals for UV-completions of d-dimensional bulk quantum gravity were recently studied in [1] by assuming that they satisfy axioms of finiteness, reality, continuity, reflection-positivity, and factorization. Sectors HB\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{H}}_{\\mathcal{B}} $$\\end{document} of the resulting Hilbert space were then defined for any (d − 2)-dimensional surface B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{B} $$\\end{document}, where B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{B} $$\\end{document} may be thought of as the boundary ∂Σ of a bulk Cauchy surface in a corresponding Lorentzian description, and where B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{B} $$\\end{document} includes the specification of appropriate boundary conditions for bulk fields. Cases where B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{B} $$\\end{document} was the disjoint union B ⊔ B of two identical (d − 2)-dimensional surfaces B were studied in detail and, after the inclusion of finite-dimensional ‘hidden sectors,’ were shown to provide a Hilbert space interpretation of the associated Ryu-Takayanagi entropy. The analysis was performed by constructing type-I von Neumann algebras ALB\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{A}}_L^B $$\\end{document}, ARB\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{A}}_R^B $$\\end{document} that act respectively at the left and right copy of B in B ⊔ B.Below, we consider the case of general B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{B} $$\\end{document}, and in particular for B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{B} $$\\end{document} = BL ⊔ BR with BL, BR distinct. For any BR, we find that the von Neumann algebra at BL acting on the off-diagonal Hilbert space sector HBL⊔BR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{H}}_{B_L\\bigsqcup {B}_R} $$\\end{document} is a central projection of the corresponding type-I von Neumann algebra on the ‘diagonal’ Hilbert space HBL⊔BL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{H}}_{B_L\\bigsqcup {B}_L} $$\\end{document}. As a result, the von Neumann algebras ALBL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{A}}_L^{B_L} $$\\end{document}, ARBL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{A}}_R^{B_L} $$\\end{document} defined in [1] using the diagonal Hilbert space HBL⊔BL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{H}}_{B_L\\bigsqcup {B}_L} $$\\end{document} turn out to coincide precisely with the analogous algebras defined using the full Hilbert space of the theory (including all sectors HB\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{H}}_{\\mathcal{B}} $$\\end{document}). A second implication is that, for any HBL⊔BR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{H}}_{B_L\\bigsqcup {B}_R} $$\\end{document}, including the same hidden sectors as in the diagonal case again provides a Hilbert space interpretation of the Ryu-Takayanagi entropy. We also show the above central projections to satisfy consistency conditions that lead to a universal central algebra relevant to all choices of BL and BR.

Axialgravisolitons at infinite corner

Abstract Gravitational solitons (gravisolitons) are particular exact solutions of Einstein field equation in vacuum build on a given background solution. Their interpretation is not yet fully clear but they contain many of the physically relevant solutions low N-solitons solutions. 
However, a systematic study and characterization of gravisolitons solution for every N is lacking and their relevance in a theory of quantum gravity is not fully understood. This work aims to investigate and characterize some properties of N-axialsoliton solutions such as their asymptotically behaviour and asymptotic symmetries given minimal assumptions on the background metric. We develop an explicit systematic asymptotically expansion for the N-axialsoliton solution and we compute the leading order of the asymptotic killing vectors. Moreover, in the perspective to better understand the role of gravisolitons in quantum gravity we make a link, and a one of the first explicit test, to the corner symmetry proposal deriving which subalgebra of the universal corner symmetry algebra is generated by the asymptotic Killing vectors of N-axialsoliton solution. In the spirit of the corner proposal, the axialgravisoliton corner symmetry algebra (agcsa) can be useful for the quantization of the non-asymptotically flat sector of gravity while, in the spirit of IR triangle, new soft theorems and memory effects could emerge.

Logical specifications of effectively separable data models

It is established that any effectively separable many-sorted universal algebra has an enrichment that is the only (up to isomorphism) model constructed from constants for a suitable computably enumerable set of sentences

Inverse homomorphisms of finite groups

Abstract Replacing the equality in the condition of the homomorphism of a multi-based universal algebra with inequality leads to the definition of an inverse homomorphism of a multi-based universal algebra. In this paper methods for constructing nontrivial inverse homomorphisms of finite groups are studied.

Boolean proportions

The author has recently introduced an abstract algebraic framework of analogical proportions within the general setting of universal algebra. This paper studies analogical proportions in the boolean domain consisting of two elements 0 and 1 within his framework. It turns out that our notion of boolean proportions coincides with two prominent models from the literature in different settings. This means that we can capture two separate modellings of boolean proportions within a single framework which is mathematically appealing and provides further evidence for the robustness and applicability of the general framework.

Wold decomposition and C*-envelopes of self-similar semigroup actions on graphs

We study the Wold decomposition for representations of a self-similar semigroup P action on a directed graph E. We then apply this decomposition to the case where P=N to study the C*-envelope of the associated universal non-selfadjoint operator algebra AN,E by carefully constructing explicit non-trivial dilations for non-boundary representations. In particular, it is shown that the C*-envelope of AN,E coincides with the self-similar graph C*-algebra OZ,E.

Free resolutions for free unitary quantum groups and universal cosovereign Hopf algebras

AbstractWe find a finite free resolution of the counit of the free unitary quantum groups of van Daele and Wang and, more generally, Bichon's universal cosovereign Hopf algebras with a generic parameter matrix. This allows us to compute Hochschild cohomology with one‐dimensional coefficients for all these Hopf algebras. In fact, the resolutions can be endowed with a Yetter–Drinfeld structure. General results of Bichon then allow us to compute also the corresponding bialgebra cohomologies. Finding the resolution rests on two pillars. We take as a starting point the resolution for the free orthogonal quantum group presented by Collins, Härtel, and Thom or its algebraic generalization to quantum symmetry groups of bilinear forms due to Bichon. Then, we make use of the fact that the free unitary quantum groups and some of its non‐Kac versions can be realized as a glued free product of a (non‐Kac) free orthogonal quantum group with , the finite group of order 2. To obtain the resolution also for more general universal cosovereign Hopf algebras, we extend Gromada's proof from compact quantum groups to the framework of matrix Hopf algebras. As a by‐product of this approach, we also obtain a projective resolution for the freely modified bistochastic quantum groups. Only a special subclass of free unitary quantum groups and universal cosovereign Hopf algebras decompose as a glued free product in the described way. In order to verify that the sequence we found is a free resolution in general (as long as the parameter matrix is generic, two conditions which are automatically fulfilled in the free unitary quantum group case), we use the theory of Hopf bi‐Galois objects and Bichon's results on monoidal equivalences between the categories of Yetter–Drinfeld modules over universal cosovereign Hopf algebras for different parameter matrices.

Automorphisms of categories of free algebras with unit for classical varieties of non-associative algebras

The goal of this paper is to describe the group of strongly stable automorphisms for categories of free finitely generated non-associative algebras with unit for classical varieties of non-associative algebras, in particular for the varieties of commutative, Jordan, alternative and power associative algebras. The motivation of this research is tightly connected with some problems in Universal Algebraic Geometry.

Conservative algebras of 2-dimensional algebras, IV

The notion of conservative algebras appeared in a paper by Kantor in 1972. Later, he defined the conservative algebra [Formula: see text] of all algebras (i.e. bilinear maps) on the [Formula: see text]-dimensional vector space. If [Formula: see text], then the algebra [Formula: see text] does not belong to any well-known class of algebras (such as associative, Lie, Jordan, or Leibniz algebras). It looks like [Formula: see text] in the theory of conservative algebras plays a similar role to the role of [Formula: see text] in the theory of Lie algebras. Namely, an arbitrary conservative algebra can be obtained from a universal algebra [Formula: see text] for some [Formula: see text] The present paper is part of a series of papers, which is dedicated to the study of the algebra [Formula: see text] and its principal subalgebras.

A Theory of Secure and Efficient Implementation of Electronic Money

Scalable and secure implementation of central bank digital currencies (CBDC) has been a challenge. Blockchains provide high operator-independent security and enable central banks to outsource CBDC operations while retaining control over the amount of circulating money. Scalability of blockchain depends on the possibility of decomposing the blockchain. We study how the choice of money scheme: accounts, bills, or unspent transaction outputs (UTXOs) influences the existence of secure and decomposable blockchain implementations of CBDC. We give formal definitions to money schemes, their decompositions, atomic decompositions inspired from the properties of blockchain implementations. For our formalism, we use tools from universal algebra and category theory. We present a general decomposition theory and conditions under which money schemes have atomic decompositions. Bill money schemes meet these conditions but account and UTXO schemes do not. Bill schemes enable scalable and secure implementations of CBDC while the more traditional schemes have some issues.

Congruences and subdirect decompositions in universal algebra

In this paper, we will focus on the topic of congruences and factorizations in universal algebra. We will first provide an introduction to universal algebra by defining lattices, algebras, congruences and other core structures. Next, we will explore the congruence and factorization properties of an algebra. Then, Birkhoff theorem indicates that every algebra can be embedded into a product of subdirectly irreducible algebras. Based on these fundamental concepts, Heyting algebra will be discussed as a typical example in universal algebra. The one-to-one correspondence between filters and congruences of a Heyting algebra will be proved. Lastly, we will show a specific way to justify the subdirectly irreducibility of a Heyting algebra.

Graph Algebras and Derived Graph Operations

We revise our former definition of graph operations and correspondingly adapt the construction of graph term algebras. As a first contribution to a prospective research field, Universal Graph Algebra, we generalize some basic concepts and results from algebras to graph algebras. To tackle this generalization task, we revise and reformulate traditional set-theoretic definitions, constructions and proofs in Universal Algebra by means of more category-theoretic concepts and constructions. In particular, we generalize the concept of generated subalgebra and prove that all monomorphic homomorphisms between graph algebras are regular. Derived graph operations are the other main topic. After an in-depth analysis of terms as representations of derived operations in traditional algebras, we identify three basic mechanisms to construct new graph operations out of given ones: parallel composition, instantiation, and sequential composition. As a counterpart of terms, we introduce graph operation expressions with a structure as close as possible to the structure of terms. We show that the three mechanisms allow us to construct, for any graph operation expression, a corresponding derived graph operation in any graph algebra.

Universal constructions for Poisson algebras. Applications

We introduce the universal algebra of two Poisson algebras P and Q as a commutative algebra A:=P(P,Q) satisfying a certain universal property. The universal algebra is shown to exist for any finite dimensional Poisson algebra P and several of its applications are highlighted. For any Poisson P-module U, we construct a functor U⊗−:AM→QPM from the category of A-modules to the category of Poisson Q-modules which has a left adjoint whenever U is finite dimensional. Similarly, if V is an A-module, then there exists another functor −⊗V:PPM→QPM connecting the categories of Poisson representations of P and Q and the latter functor also admits a left adjoint if V is finite dimensional. If P is n-dimensional, then P(P):=P(P,P) is the initial object in the category of all commutative bialgebras coacting on P. As an algebra, P(P) can be described as the quotient of the polynomial algebra k[Xij|i,j=1,⋯,n] through an ideal generated by 2n3 non-homogeneous polynomials of degree ≤2. Two applications are provided. The first one describes the automorphisms group AutPoiss(P) as the group of all invertible group-like elements of the finite dual P(P)o. Secondly, we show that for an abelian group G, all G-gradings on P can be explicitly described and classified in terms of the universal coacting bialgebra P(P).

Combinatorial relations among relations for level 2 standard Cn(1)-modules

For an affine Lie algebra ĝ, the coefficients of certain vertex operators that annihilate level k standard ĝ-modules are the defining relations for level k standard modules. In this paper, we study a combinatorial structure of the leading terms of these relations for level k = 2 standard ĝ-modules for affine Lie algebras of type Cn(1) and the main result is the construction of combinatorially parameterized relations among the coefficients of annihilating fields. It is believed that the constructed relations among relations will play a key role in the construction of Gröbner-like basis of the maximal ideal of the universal vertex operator algebra Vgk for k = 2.

ALGEBRAIC CHARACTERISTICS OF THE CRITERION OF COMPLETENESS OF A CLASS OF ALGEBRAIC SYSTEMS

In many sources on model theory, in addition to the proven properties about classes of algebraic systems, the characteristics of these properties are given in algebraic terms, that is, they show the nature of these properties from the perspective of universal algebra. For example, the class of quasi-varieties or varieties is defined using model-theoretic concepts, fulfillment of quasi-identities or identities, and in algebraic concepts of closedness with respect to direct products, ultraproducts, fulfillment of locality, closedness with respect to homomorphisms. H.J. Keisler gave an algebraic characterization of the criterion for the axiomatizability of a class of algebraic systems, using the closure of the class under the ultraproduct and isomorphism of algebraic systems, as well as the closure under ultrapowers to complement the class. H.J. Keisler, however, does not give any algebraic characterization of the criterion for completeness of a class of algebraic systems.In this article, an algebraic characterization of the completeness criterion for a class of algebraic systems is obtained. For comparison, it is not possible to give an algebraic description of the criterion for the model completeness of a class, in terms used in the article. This shows that the algebraic nature of complete and model complete classes is somewhat different.