Integrable deformations of sigma models

In this pedagogical review we introduce systematic approaches to deforming integrable 2-dimensional sigma models. We use the integrable principal chiral model and the conformal Wess-Zumino-Witten model as our starting points and explore their Yang-Baxter and current-current deformations. There is an intricate web of relations between these models based on underlying algebraic structures and worldsheet dualities, which is highlighted throughout. We finish with a discussion of the generalisation to other symmetric integrable models, including some original results related to ZT cosets and their deformations, and the application to string theory. This review is based on notes written for lectures delivered at the school "Integrability, Dualities and Deformations," which ran from 23 to 27 August 2021 in Santiago de Compostela and virtually.

Confluence of algebraic rewriting systems

Convergent rewriting systems on algebraic structures give methods to solve decision problems, to prove coherence results, and to compute homological invariants. These methods are based on higher-dimensional extensions of the critical branching lemma that proves local confluence from confluence of the critical branchings. The analysis of local confluence of rewriting systems on algebraic structures, such as groups or linear algebras, is complicated because of the underlying algebraic axioms. This article introduces the structure of algebraic polygraph modulo that formalizes the interaction between the rules of an algebraic rewriting system and the inherent algebraic axioms, and we show a critical branching lemma for algebraic polygraphs. We deduce a critical branching lemma for rewriting systems on algebraic models whose axioms are specified by convergent modulo rewriting systems. We illustrate our constructions for string, linear, and group rewriting systems.

Multiplicative Dirac System

We define a Dirac system in multiplicative calculus by some algebraic structures. Asymptotic estimates for eigenfunctions of the multiplicative Dirac system are obtained. Eventually, some fundamental properties of the multiplicative Dirac system are examined in detail.

Matching BiHom-Rota-Baxter Algebras and Related Structures

In this paper, we introduce the notions of matching BiHom-Rota-Baxter algebras, matching BiHom-(tri)dendriform algebras, matching BiHom-Zinbiel algebras and matching BiHom-pre-Lie algebras. Moreover, we study the properties and relationships between categories of these matching BiHom-algebraic structures.

Tensor Hierarchy Algebra Extensions of Over-Extended Kac–Moody Algebras

Tensor hierarchy algebras are infinite-dimensional generalisations of Cartan-type Lie superalgebras. They are not contragredient, exhibiting an asymmetry between positive and negative levels. These superalgebras have been a focus of attention due to the fundamental rôle they play for extended geometry. In the present paper, we examine tensor hierarchy algebras which are super-extensions of over-extended (often, hyperbolic) Kac–Moody algebras. They contain novel algebraic structures. Of particular interest is the extension of a over-extended algebra by its fundamental module, an extension that contains and generalises the extension of an affine Kac–Moody algebra by a Virasoro derivation $$L_1$$ L 1 . A conjecture about the complete superalgebra is formulated, relating it to the corresponding Borcherds superalgebra.

Logical Reasoning for Forecasting in Mesopotamia

In this paper, I show that a kind of perfect logical competence is observed in the Babylonian tablets used for forecasting. In these documents, we see an intuition of some algebraic structures that are used for inferring prognoses as logical conclusions. The paper is based mainly on the omen series reconstructed by N. De Zorzi. It is shown that in composing these divination lists there was implicitly used the Boolean algebra.

On Neutrosophic Quadruple Groups

As generalizations and alternatives of classical algebraic structures there have been introduced in 2019 the NeutroAlgebraic structures (or NeutroAlgebras) and AntiAlgebraic structures (or AntiAlgebras). Unlike the classical algebraic structures, where all operations are well defined and all axioms are totally true, in NeutroAlgebras and AntiAlgebras, the operations may be partially well defined and the axioms partially true or, respectively, totally outer-defined and the axioms totally false. These NeutroAlgebras and AntiAlgebras form a new field of research, which is inspired from our real world. In this paper, we study neutrosophic quadruple algebraic structures and NeutroQuadrupleAlgebraicStructures. NeutroQuadrupleGroup is studied in particular and several examples are provided. It is shown that $$(NQ({\mathbb {Z}}),\div )$$ ( N Q ( Z ) , ÷ ) is a NeutroQuadrupleGroup. Substructures of NeutroQuadrupleGroups are also presented with examples.

An Application of Sombor Index over a Special Class of Semigroup Graph

Recently, Gutman introduced a class of novel topological invariants named Sombor index which is defined as S O G = ∑ u v ∈ E G d u 2 + d v 2 . In this study, the Sombor index of monogenic semigroup graphs, which is an important class of algebraic structures, is calculated.