In this paper, we first introduce embedding tensors on a Hom-Lie algebra with respect to a given representation. An embedding tensor naturally induces a Hom-Leibniz algebra structure. We construct a graded Lie algebra that characterizes embedding tensors as its Maurer-Cartan elements. Using this, we define the cohomology of an embedding tensor and realize it as the cohomology of the induced Hom-Leibniz algebra with coefficients in a suitable representation. A triple consisting of a Hom-Lie algebra, a representation and an embedding tensor is called a Hom-Lie-Leibniz triple. We construct the controlling L ∞ -algebra of a given Hom-Lie-Leibniz triple. Next, we define the cohomology of a Hom-Lie-Leibniz triple that governs the deformations of the structure. Finally, we introduce homotopy embedding tensors, HLeib ∞ -algebras and find their relations.

Neural network quantization is an established technique for compressing real-valued models, but its application to complex-valued networks-essential in electromagnetics, acoustics, and quantum physics-remains underdeveloped. Conventional quantization methods treat real and imaginary components as independent channels, thereby disrupting the algebraic structure of complex multiplication and distorting essential phase relationships. To address this problem, we propose a real-imaginary joint quantization method that minimizes error propagation in complex multiplication and maintains coherence in phase-sensitive tasks, thereby preserving amplitude-phase fidelity during complex-valued inference. Combined with physics-aware adaptive precision training, this approach demonstrates outstanding performance across hologram generation, audio classification, wireless signal classification, and synthetic aperture radar signal recognition tasks. Compared to the state-of-the-art hologram generation model HoloNet, our approach achieves a 3.9 dB improvement in peak signal-to-noise ratio while reducing computational load and memory consumption by 99.1% and 99.8%, respectively. This research establishes a pathway toward lightweight, high-fidelity complex-valued neural networks for scientific computing and coherent signal processing.

Abstract In this note, we determine, by a disjunctive normal form theorem, which functions on the standard n -nuanced Łukasiewicz-Moisil algebra are representable by formulas and we show how this result may help in establishing the structure of the free algebras in this class.

In this work, we systematically treat the ambiguities that generically arise in the gradient expansion of any hydrodynamic theory. While these ambiguities do not affect the physical content of the equations, they induce two types of transformations in the space of transport coefficients. The first type is known as the “frame” transformations, and amounts to field redefinitions. The second type, which we introduce and formalize here, we term the “on-shell” transformations. This identifies equivalence classes of hydrodynamic theories that provide an equally valid low-energy description of the underlying microscopic theory. We show that in any (classical) theory of hydrodynamics (at arbitrary order in derivatives), the action of such transformations on the dispersion relations and two-point correlation functions is universal. We explicitly construct invariants which can then be matched to a microscopic theory. Among them are, expectedly, the low-momentum expansions of the hydrodynamic modes. The (unphysical) gapped modes can, however, be added or removed at will. Finally, we show that such transformations assign a nilpotent Lie group to every hydrodynamic theory, and discuss the related algebraic properties underlying classical hydrodynamics.

We characterize weak compactness in the Sobolev space Wk,∞(Ω). For non-reflexive spaces like Wk,∞, criteria beyond boundedness are required. By exploiting the von Neumann algebra structure of L∞ via Gelfand duality, we establish a unified theory. Our main result is a necessary and sufficient condition: a subset is relatively weakly compact if and only if it is bounded and its weak derivatives up to order k have uniformly small oscillation on a finite measurable partition of Ω. This provides a tool for analyzing nonlinear problems in these spaces.

Accurately modeling quantum dissipative dynamics remains challenging due to environmental complexity and non-Markovian memory effects. Although machine learning provides a promising alternative to conventional simulation techniques, most existing models employ real-valued neural networks (RVNNs) that inherently mismatch the complex-valued nature of quantum mechanics. By decoupling the real and imaginary parts of the density matrix, RVNNs can obscure essential amplitude-phase correlations, compromising physical consistency. Here, we introduce complex-valued neural networks (CVNNs) as a physics-consistent framework for learning quantum dissipative dynamics. CVNNs operate directly on complex-valued inputs, preserve the algebraic structure of quantum states, and naturally encode quantum coherences. Through numerical benchmarks on the spin-boson model and few variants of the Fenna-Matthews-Olson complex, we demonstrate that CVNNs outperform RVNNs in convergence speed, training stability, and physical fidelity-including significantly improved trace conservation and Hermiticity. These advantages increase with system size and coherence complexity, establishing CVNNs as a robust, scalable, quantum-aware classical approach for simulating open quantum systems in the pre-fault-tolerant quantum era.

Abstract We introduce a framework to define coalgebra and bialgebra structures on two-dimensional (2D) square lattices, extending the algebraic theory of Hopf algebras and quantum groups beyond the one-dimensional (1D) setting. Our construction is based on defining 2D coproducts through horizontal and vertical maps that satisfy compatibility and associativity conditions, enabling the consistent growth of vector spaces over lattice sites. We present several examples of 2D bialgebras, including group-like and Lie algebra-inspired constructions and a quasi-1D coproduct instance that is applicable to Taft-Hopf algebras and to quantum groups. The approach is further applied to the quantum group $U_q[su(2)]$, for which we construct 2D generalizations of its generators, analyze $q$-deformed singlet states, and derive a 2D R-matrix satisfying an intertwining relation in the semiclassical limit. Additionally, we show how tensor network states, particularly PEPS, naturally induce 2D coalgebra structures when supplemented with appropriate boundary conditions. Our results establish a local and algebraically consistent method to embed quantum group symmetries into higher-dimensional lattice systems, potentially connecting to the emerging theory of fusion 2-categories and categorical symmetries in quantum many-body physics.

UDC 512.5 We focus on recent promising trends in the application of the key concepts and approaches from the classical infinite-group theory to various branches of algebra, such as modules over group rings, infinite-dimensional linear groups, Leibniz algebras, other generalizations of the Lie algebras, and braces. The efficacy of these trends has been well-documented in a series of recent books from reputable publishers. In our article, we present a concise overview of these emerging trends. The analysis of the mutual influence of algebraic systems promotes deeper understanding of their individual and collective significance and illustrates the unity and diversity typical of contemporary mathematics. We believe that the subsequent development of investigations in this field would promote the appearance of new discoveries and innovations clarifying the fundamental role played by the groups, rings, algebras and other algebraic structures in modern mathematics.

In this paper, we describe explicitly the structure of the derivation algebra and automorphism group of the symplectic oscillator Lie algebra [Formula: see text] ([Formula: see text]), where [Formula: see text] is the symplectic Lie algebra and [Formula: see text] is the [Formula: see text]-dimensional Heisenberg algebra.

Let [Formula: see text] be a class of non-finitely graded Lie algebras related to generalized Witt algebras [Formula: see text]. We give a full description of the Poisson structures on [Formula: see text]. It is shown that there is no non-trivial and associative Poisson algebraic structures on [Formula: see text].

Classical cyclic and constacyclic codes play a fundamental role in coding theory thanks to their rich algebraic structures and efficient encoding/decoding algorithms. These codes are especially important in the construction of quantum error-correcting codes, since their inherent algebraic properties facilitate the derivation of self-orthogonality or dual-containing conditions that are crucial for ensuring quantum error correction capability. In this paper, we construct entanglement-assisted quantum error-correcting codes (EAQEC codes) of length 5p t over 𝔽 p m . We compare our EAQEC codes with all known EAQEC codes to see that our EAQEC codes are new in the sense that their parameters are different from all the previous constructions. Moreover, our EAQEC codes have shorter lengths than the known codes, yet exhibit significantly larger quantum distances. This is highly meaningful in the context of quantum error correction, as it enables the construction of codes with small lengths while maintaining large quantum distances. We also construct asymmetric entanglement-assisted quantum error-correcting (AEAQEC) codes of length 5p t over 𝔽 p m . We evaluate our AEAQEC codes against all existing ones and find that they are both new and better.

In this paper, we introduce the notion of modified Rota-Baxter operators of non-zero weight on [Formula: see text]-Lie algebras and provide some examples. Next, we give various constructions of modified Rota-Baxter operators of non-zero weight according to constructions of [Formula: see text]-Lie algebras. Furthermore, we define a cohomology of modified Rota-Baxter operators of non-zero weight on [Formula: see text]-Lie algebras with coefficients in a suitable representation. As an application, we study formal deformations of modified Rota-Baxter operators of non-zero weight that are generated by the above-defined cohomology. In the final part of the paper, we construct two [Formula: see text]-algebra structures whose Maurer-Cartan elements correspond to relative and absolute modified Rota-Baxter [Formula: see text]-Lie algebra structures of nonzero weight, respectively. Lastly, we compare our [Formula: see text]-algebraic approach with the deformation-controlling [Formula: see text]-algebra for relative Rota-Baxter [Formula: see text]-Lie operators developed by Hou, Sheng, and Zhou.

In this paper, we introduce and develop the notions of vague topological ring, vague topological field, vague topological module, and vague topological vector space within the framework of vague set theory. By combining the algebraic structures of rings, fields, and modules with vague topology, we establish a generalised setting that extends classical topological algebra. We present fundamental definitions, illustrative examples, and examine important structural properties of these newly defined concepts. Furthermore, we investigate the relationships between vague topological structures and their corresponding classical counterparts, and analyse continuity conditions of algebraic operations under vague topology. The results obtained contribute to the advancement of vague algebraic topology and provide a foundation for further research in this direction.

Let [Formula: see text] be the Leavitt path algebra of a directed graph [Formula: see text] over a field [Formula: see text]. In this paper, we determine [Formula: see text] and [Formula: see text] for the Lie algebra [Formula: see text] and the Jordan algebra [Formula: see text] arising from [Formula: see text] with respect to the standard involution to be solvable.

This paper introduces the notion of an anti-symmetric covariant (ASC) bialgebra as a generalization of a balanced infinitesimal bialgebra. Our primary focus is on the construction of these algebraic structures from associative Yang–Baxter pairs (AYBPs). We demonstrate that this construction leads to a quasi-triangular ASC bialgebra under the conditions that the AYBP is [Formula: see text]-invariant and [Formula: see text]-symmetric. The relationship between AYBPs in symmetric Frobenius algebras and Rota–Baxter systems is investigated using the formalism of [Formula: see text]-operator systems. A central theme of this work is the development of a factorization theory for a special class of these structures, termed factorizable ASC bialgebras. We prove that a factorizable ASC bialgebra induces a factorization of its underlying associative algebra. To characterize these structures, we introduce the concept of a symmetric Rota–Baxter Frobenius algebra and establish a one-to-one correspondence between them and factorizable ASC bialgebras. This correspondence is subsequently employed to construct examples of factorizable ASC bialgebras from Rota–Baxter systems. Finally, we present an algorithm for computing AYBPs in finite-dimensional associative algebras and apply it to classify all such pairs in two-dimensional complex associative algebras, yielding concrete examples of the aforementioned structures.

We study the algebraic structure of one-loop Bern-Carrasco-Johansson numerators in Yang-Mills and related theories. Starting from the propagator matrix that connects color-ordered integrands to numerators, we identify the consistency conditions that ensure the existence of Jacobi-satisfying numerator solutions and determine the unique construction. The relation between one-loop numerators and forward-limit tree numerators is clarified, together with the additional physical conditions required for a consistent double-copy interpretation. We propose a two-step expansion strategy for obtaining explicit one-loop numerators. The Yang-Mills integrand is first decomposed into scalar-loop Yang-Mills-scalar building blocks, which are then expanded into biadjoint scalar integrands. We derive explicit results for up to three external gluons, showing how the kinematic consistency conditions uniquely determine the coefficients in each case. Similar results for Einstein-Yang-Mills and gravity amplitudes are also presented.

We present a framework for the formal modeling of state-based systems in the context of the Lean4 programming language and proof assistant. In this context, the main objective is to support a step-wise refinement methodology inspired conceptually by the Event-B formal method. As a starting point, the LeanMachines framework proposes Lean4 constructions for the main Event-B concepts such as contexts, machines and events. Most importantly, the associated refinement principles are introduced in the form of typeclass constructions inspired by (and in fact built upon) the Mathlib mathematical framework. Beyond the basic concepts and structures, we also experiment with extensions of the framework. First, we develop an algebra of event combinators that allow to compose complex event structures out of simpler ones. These combinators are based on algebraic structures – functors, arrows, etc. – that have been developed and studied in the context of (functional) programming language theory. Our proposed formalization of the Event-B concepts is very shallow in the sense that all the constructions are directly based on the Lean4 logic and abstractions. One benefit is that proof obligations can be discharged using the tactic language of Lean4 with almost no embedding overhead such as an abstraction barrier that would require syntactic conversions, or the necessity to use some dedicated proof tactics. As an important design guideline, we enforce the fundamental principle of correctness-by-construction: machine states, events structures and refinement steps cannot be fully constructed without discharging the prescribed proof obligations.

Abstract We complete the classification of algebraic monoid structures on the affine 3-space. The result is based on a reduction of the general case to that of commutative monoids. We also study various algebraic properties of all monoids appearing in the classification.

Analyzing the structure of the automorphism groups of graphs, and investigating the properties of graphs that are constructed from algebraic structures are two important research topics in algebraic graph theory. Blending these two aspects of study, an algebraic intersection graph, called the invariant intersection graph of a graph, has been introduced in the literature. In this paper, we study certain properties of the invariant intersection graphs of graphs, and obtain some structural characterizations of these graphs, based on the automorphism group of the graph on which the invariant intersection graph is constructed.

Physical-layer security (PLS) provides an information-theoretic framework for securing wireless communications by exploiting channel and signal-structure asymmetries, thereby avoiding reliance on computational hardness assumptions. Within this setting, lattice codes and their algebraic constructions play a central role in achieving secrecy over Gaussian and fading wiretap channels. This article offers a comprehensive survey of lattice-based wiretap coding, covering foundational concepts in algebraic number theory, Construction A over number fields, and the structure of modular and unimodular lattice families. We review key secrecy metrics, including secrecy gain, flatness factor, and equivocation, and consolidate classical and recent results to provide a unified perspective that links wireless-channel models with their underlying algebraic lattice structures. In addition, we review a newly proposed family of p-modular lattices in Khodaiemehr, H., 2018 constructed from cyclotomic fields Q(ζp) for primes p≡1(mod4) via a generalized Construction A framework. We characterize their algebraic and geometric properties and establish a non-existence theorem showing that such constructions cannot be extended to prime-power cyclotomic fields Q(ζpn) with n>1. Finally, motivated by the fact that these p-modular lattices naturally yield mixed-signature structures for which classical theta series diverge, we integrate recent advances on indefinite theta series and modular completions. Drawing on Vignéras' differential framework and generalized error functions, we outline how modularly completed indefinite theta series provide a principled analytic foundation for defining secrecy-relevant quantities in the indefinite setting. Overall, this work serves both as a survey of algebraic lattice techniques for PLS and as a source of new design insights for secure wireless communication systems.