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21 pages, 315 KiB

Open AccessArticle

Module Algebra Structures of Non-Standard Quantum Group X_q(A₁) on

C

[x,y,z]

by Dong Su

Mathematics 2025, 13(14), 2227; https://doi.org/10.3390/math13142227 - 8 Jul 2025

Viewed by 256

Abstract

In this paper, we investigate the module algebra structures of

X_{q} (A_{1})

on the quantum polynomial algebra

C_{q} [x, y, z]

. The automorphism group of [...] Read more.

In this paper, we investigate the module algebra structures of

X_{q} (A_{1})

on the quantum polynomial algebra

C_{q} [x, y, z]

. The automorphism group of

C_{q} [x, y, z]

is denoted by

Aut (C_{q} [x, y, z])

, and we give a complete classification of

X_{q} (A_{1})

-module algebra structures on

C_{q} [x, y, z]

, where

K_{1}, K_{2} \in Aut (C_{q} [x, y, z])

. Full article

28 pages, 847 KiB

Open AccessArticle

The Standard Model Symmetry and Qubit Entanglement

by Jochen Szangolies

Entropy 2025, 27(6), 569; https://doi.org/10.3390/e27060569 - 27 May 2025

Viewed by 921

Abstract

Research at the intersection of quantum gravity and quantum information theory has seen significant success in describing the emergence of spacetime and gravity from quantum states whose entanglement entropy approximately obeys an area law. In a different direction, the Kaluza–Klein proposal aims to [...] Read more.

Research at the intersection of quantum gravity and quantum information theory has seen significant success in describing the emergence of spacetime and gravity from quantum states whose entanglement entropy approximately obeys an area law. In a different direction, the Kaluza–Klein proposal aims to recover gauge symmetries by means of dimensional reduction in higher-dimensional gravitational theories. Integrating both of these, gravitational and gauge degrees of freedom in

3 + 1

dimensions may be obtained upon dimensional reduction in higher-dimensional emergent gravity. To this end, we show that entangled systems of two and three qubits can be associated with

5 + 1

- and

9 + 1

-dimensional spacetimes, respectively, which are reduced to

3 + 1

dimensions upon singling out a preferred complex direction. Depending on the interpretation of the residual symmetry, either the Standard Model gauge group,

S U (3) \times S U (2) \times U (1) / Z_{6}

, or the symmetry of Minkowski spacetime together with the gauge symmetry of a right-handed ‘half-generation’ of fermions can be recovered. Thus, there seems to be a natural way to accommodate the chirality of the weak force in the given construction. This motivates a picture in which spacetime emerges from the area law contribution to the entanglement entropy, while gauge and matter degrees of freedom are obtained due to area-law-violating terms. Furthermore, we highlight the possibility of using this construction in quantum simulations of Standard Model fields. Full article

(This article belongs to the Special Issue Foundational Aspects of Gauge Field Theory)

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12 pages, 222 KiB

Open AccessArticle

Weak Hopf Algebra Structures on Hybrid Numbers

by Jiangang Tang and Quanguo Chen

Symmetry 2025, 17(6), 828; https://doi.org/10.3390/sym17060828 - 26 May 2025

Viewed by 318

Abstract

Let

K

be the algebra of hybrid numbers. In this paper, a weak Hopf algebra structure is endowed on

K

. Additionally, the integrals in this weak Hopf algebra

K

are discussed. Full article

(This article belongs to the Section Mathematics)

15 pages, 283 KiB

Open AccessArticle

A Class of Non-Hopf Bi-Frobenius Algebras Generated by n Elements

by Zhan Fa and Yanhua Wang

Mathematics 2025, 13(8), 1357; https://doi.org/10.3390/math13081357 - 21 Apr 2025

Viewed by 222

Abstract

Bi-Frobenius algebras are a class of Frobenius algebras and Frobenius coalgebras with some compatible conditions. In this paper, we construct a class of bi-Frobenius algebras generated by n elements on graded algebra

A .

The comultiplication and counit are defined via a permutation [...] Read more.

Bi-Frobenius algebras are a class of Frobenius algebras and Frobenius coalgebras with some compatible conditions. In this paper, we construct a class of bi-Frobenius algebras generated by n elements on graded algebra

A .

The comultiplication and counit are defined via a permutation

π

on

A,

such that A becomes a bi-Frobenius algebra. For any n, these bi-Frobenius algebras are neither Hopf algebras nor S-type bi-Frobenius algebras. Full article

13 pages, 236 KiB

Open AccessArticle

Multiplier Left Hopf Algebras

by Chunxiao Yan and Shuanhong Wang

Mathematics 2025, 13(7), 1138; https://doi.org/10.3390/math13071138 - 30 Mar 2025

Viewed by 282

Abstract

In this paper, we introduce and study the notion of a multiplier left Hopf algebra, which can be seen as an extension of the Van Daele’s multiplier Hopf algebras and the Green–Nichols–Taft’s left Hopf algebras. In particular, we investigate the relation between the [...] Read more.

In this paper, we introduce and study the notion of a multiplier left Hopf algebra, which can be seen as an extension of the Van Daele’s multiplier Hopf algebras and the Green–Nichols–Taft’s left Hopf algebras. In particular, we investigate the relation between the notion of the Van Daele’s left multiplier Hopf algebras and the one of our multiplier left Hopf algebras. Finally, we determine the case when a multiplier left Hopf algebra becomes a multiplier Hopf algebra. Full article

12 pages, 270 KiB

Open AccessArticle

Total Momentum and Other Noether Charges for Particles Interacting in a Quantum Spacetime

by Giovanni Amelino-Camelia, Giuseppe Fabiano and Domenico Frattulillo

Symmetry 2025, 17(2), 227; https://doi.org/10.3390/sym17020227 - 5 Feb 2025

Cited by 4 | Viewed by 502

Abstract

There has been strong interest in the fate of relativistic symmetries in some quantum spacetimes, partly because of its possible relevance for high-precision experimental tests of relativistic properties. However, the main technical results obtained so far concern the description of suitably deformed relativistic [...] Read more.

There has been strong interest in the fate of relativistic symmetries in some quantum spacetimes, partly because of its possible relevance for high-precision experimental tests of relativistic properties. However, the main technical results obtained so far concern the description of suitably deformed relativistic symmetry transformation rules, whereas the properties of the associated Noether charges, which are crucial for the phenomenology, are still poorly understood. Here, we tackle this problem focusing on first-quantized particles described within a Hamiltonian framework and using as a toy model the so-called “spatial kappa-Minkowski noncommutative spacetime”, where all the relevant conceptual challenges are present but, as here shown, in technically manageable fashion. We derive the Noether charges, including the much-debated total momentum charges, and we reveal a strong link between the properties of these Noether charges and the structure of the laws of interaction among particles. Full article

(This article belongs to the Section Physics)

16 pages, 268 KiB

Open AccessArticle

Bratteli Diagrams, Hopf–Galois Extensions and Calculi

by Ghaliah Alhamzi and Edwin Beggs

Symmetry 2025, 17(2), 164; https://doi.org/10.3390/sym17020164 - 22 Jan 2025

Viewed by 603

Abstract

Hopf–Galois extensions extend the idea of principal bundles to noncommutative geometry, using Hopf algebras as symmetries. We show that the matrix embeddings in Bratteli diagrams are iterated direct sums of Hopf–Galois extensions (quantum principal bundles) for certain finite abelian groups. The corresponding strong [...] Read more.

Hopf–Galois extensions extend the idea of principal bundles to noncommutative geometry, using Hopf algebras as symmetries. We show that the matrix embeddings in Bratteli diagrams are iterated direct sums of Hopf–Galois extensions (quantum principal bundles) for certain finite abelian groups. The corresponding strong universal connections are computed. We show that

M_{n} (C)

is a trivial quantum principle bundle for the Hopf algebra

C [Z_{n} \times Z_{n}]

. We conclude with an application relating calculi on groups to calculi on matrices. Full article

(This article belongs to the Section Mathematics)

17 pages, 305 KiB

Open AccessArticle

Lie Bialgebra Structures and Quantization of Generalized Loop Planar Galilean Conformal Algebra

by Yu Yang and Xingtao Wang

Axioms 2025, 14(1), 7; https://doi.org/10.3390/axioms14010007 - 26 Dec 2024

Viewed by 653

Abstract

In this paper, we analyze the Lie bialgebra (LB) and quantize the generalized loop planar-Galilean conformal algebra (GLPGCA)

W (Γ)

. Additionally, we prove that all LB structures on

W (Γ)

possess a triangular coboundary. We also quantize [...] Read more.

In this paper, we analyze the Lie bialgebra (LB) and quantize the generalized loop planar-Galilean conformal algebra (GLPGCA)

W (Γ)

. Additionally, we prove that all LB structures on

W (Γ)

possess a triangular coboundary. We also quantize

W (Γ)

using the Drinfeld-twist quantization technique and identify a group of noncommutative algebras and noncocommutative Hopf algebras. Full article

12 pages, 270 KiB

Open AccessArticle

Some Notes on Higher Frobenius–Schur Indicators of the Regular Representations for Matched Pairs of Groups

by Chenfei Mao and Kangqiao Li

Mathematics 2024, 12(22), 3479; https://doi.org/10.3390/math12223479 - 7 Nov 2024

Viewed by 898

Abstract

The notion of matched pair

(F, G, ▹, ◃)

of finite groups was introduced by Takeuchi in 1981, which is equivalent to a factorization of the group

F ⋈ G

. We find in this paper some sufficient [...] Read more.

The notion of matched pair

(F, G, ▹, ◃)

of finite groups was introduced by Takeuchi in 1981, which is equivalent to a factorization of the group

F ⋈ G

. We find in this paper some sufficient conditions when equations

ν_{m} (F ⋈ G) = ν_{m} (F) ν_{m} (G)

for all

m \geq 0

imply that

F ⋈ G

is the external direct product of

F \times G

, where

ν_{m}

denotes m-th indicator of the regular representation of a finite group. A comparison with indicators of bismash product Hopf algebras

C^{G} # C F

is also mentioned. Full article

(This article belongs to the Section A: Algebra and Logic)

23 pages, 319 KiB

Open AccessArticle

Quasigroups, Braided Hopf (Co)quasigroups and Radford’s Biproducts of Quasi-Diagonal Type

by Yue Gu and Shuanhong Wang

Mathematics 2024, 12(21), 3384; https://doi.org/10.3390/math12213384 - 29 Oct 2024

Cited by 1 | Viewed by 940

Abstract

Given the Yetter–Drinfeld category over any quasigroup and a braided Hopf coquasigroup in this category, we first mainly study the Radford’s biproduct corresponding to this braided Hopf coquasigroup. Then, we investigate Sweedler’s duality of this braided Hopf coquasigroup and show that this duality [...] Read more.

Given the Yetter–Drinfeld category over any quasigroup and a braided Hopf coquasigroup in this category, we first mainly study the Radford’s biproduct corresponding to this braided Hopf coquasigroup. Then, we investigate Sweedler’s duality of this braided Hopf coquasigroup and show that this duality is also a braided Hopf quasigroup in the Yetter–Drinfeld category, generalizing the main result in a Hopf algebra case of Ng and Taft’s paper. Finally, as an application of our results, we show that the space of binary linearly recursive sequences is closed under the quantum convolution product of binary linearly recursive sequences. Full article

16 pages, 4720 KiB

Open AccessArticle

Dynamics of a New Four-Thirds-Degree Sub-Quadratic Lorenz-like System

by Guiyao Ke, Jun Pan, Feiyu Hu and Haijun Wang

Axioms 2024, 13(9), 625; https://doi.org/10.3390/axioms13090625 - 12 Sep 2024

Cited by 4 | Viewed by 866

Abstract

Aiming to explore the subtle connection between the number of nonlinear terms in Lorenz-like systems and hidden attractors, this paper introduces a new simple sub-quadratic four-thirds-degree Lorenz-like system, where

\dot{x} = a (y - x)

, [...] Read more.

Aiming to explore the subtle connection between the number of nonlinear terms in Lorenz-like systems and hidden attractors, this paper introduces a new simple sub-quadratic four-thirds-degree Lorenz-like system, where

\dot{x} = a (y - x)

,

\dot{y} = c x - \sqrt[3]{x} z

,

\dot{z} = - b z + \sqrt[3]{x} y

, and uncovers the following property of these systems: decreasing the powers of the nonlinear terms in a quadratic Lorenz-like system where

\dot{x} = a (y - x)

,

\dot{y} = c x - x z

,

\dot{z} = - b z + x y

, may narrow, or even eliminate the range of the parameter c for hidden attractors, but enlarge it for self-excited attractors. By combining numerical simulation, stability and bifurcation theory, most of the important dynamics of the Lorenz system family are revealed, including self-excited Lorenz-like attractors, Hopf bifurcation and generic pitchfork bifurcation at the origin, singularly degenerate heteroclinic cycles, degenerate pitchfork bifurcation at non-isolated equilibria, invariant algebraic surface, heteroclinic orbits and so on. The obtained results may verify the generalization of the second part of the celebrated Hilbert’s sixteenth problem to some degree, showing that the number and mutual disposition of attractors and repellers may depend on the degree of chaotic multidimensional dynamical systems. Full article

(This article belongs to the Section Mathematical Analysis)

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16 pages, 323 KiB

Open AccessArticle

Quantization of the Rank Two Heisenberg–Virasoro Algebra

by Xue Chen

Axioms 2024, 13(7), 446; https://doi.org/10.3390/axioms13070446 - 1 Jul 2024

Viewed by 845

Abstract

Quantum groups occupy a significant position in both mathematics and physics, contributing to progress in these fields. It is interesting to obtain new quantum groups by the quantization of Lie bialgebras. In this paper, the quantization of the rank two Heisenberg–Virasoro algebra by [...] Read more.

Quantum groups occupy a significant position in both mathematics and physics, contributing to progress in these fields. It is interesting to obtain new quantum groups by the quantization of Lie bialgebras. In this paper, the quantization of the rank two Heisenberg–Virasoro algebra by Drinfel’d twists is presented, Lie bialgebra structures of which have been investigated by the authors recently. Full article

(This article belongs to the Section Algebra and Number Theory)

15 pages, 294 KiB

Open AccessArticle

The Hopf Automorphism Group of Two Classes of Drinfeld Doubles

by Hua Sun, Mi Hu and Jiawei Hu

Symmetry 2024, 16(6), 735; https://doi.org/10.3390/sym16060735 - 12 Jun 2024

Viewed by 967

Abstract

Let

D (R_{m, n} (q))

be the Drinfeld double of Radford Hopf algebra

R_{m, n} (q)

and

D (H_{s, t})

be the Drinfeld double of generalized Taft algebra [...] Read more.

Let

D (R_{m, n} (q))

be the Drinfeld double of Radford Hopf algebra

R_{m, n} (q)

and

D (H_{s, t})

be the Drinfeld double of generalized Taft algebra

H_{s, t}

. Both

D (R_{m, n} (q))

and

D (H_{s, t})

have very symmetric structures. We calculate all Hopf automorphisms of

D (R_{m, n} (q))

and

D (H_{s, t})

, respectively. Furthermore, we prove that the Hopf automorphism group

A u t_{H o p f} (D (R_{m, n} (q)))

is isomorphic to the direct sum

Z_{n} ⨁ Z_{m}

of cyclic groups

Z_{m}

and

Z_{n}

, the Hopf automorphism group

A u t_{H o p f} (D (H_{s, t}))

is isomorphic to the semi-direct products

k^{*} ⋊ Z_{d}

of multiplicative group

k^{*}

and cyclic group

Z_{d}

, where

s = t d, k^{*} = k \ {0}

, and

k

is an algebraically closed field with char

(k) ∤ t

. Full article

(This article belongs to the Section Mathematics)

12 pages, 274 KiB

Open AccessArticle

The Ribbon Elements of the Quantum Double of Generalized Taft–Hopf Algebra

by Hua Sun, Yuyan Zhang, Ziliang Jiang, Mingyu Huang and Jiawei Hu

Mathematics 2024, 12(12), 1802; https://doi.org/10.3390/math12121802 - 10 Jun 2024

Viewed by 1048

Abstract

Let s, t be two positive integers and

k

be an algebraically closed field with char (

k) ∤ s t

. We show that the Drinfeld double

D (⋀_{s t, t}^{* c o p})

of [...] Read more.

Let s, t be two positive integers and

k

be an algebraically closed field with char (

k) ∤ s t

. We show that the Drinfeld double

D (⋀_{s t, t}^{* c o p})

of generalized Taft–Hopf algebra

⋀_{s t, t}^{* c o p}

has ribbon elements if and only if t is odd. Moreover, if s is even and t is odd, then

D (⋀_{s t, t}^{* c o p})

has two ribbon elements, and if both s and t are odd, then

D (⋀_{s t, t}^{* c o p})

has only one ribbon element. Finally, we compute explicitly all ribbon elements of

D (⋀_{s t, t}^{* c o p})

. Full article

(This article belongs to the Special Issue Advanced Research in Pure and Applied Algebra)

10 pages, 250 KiB

Open AccessArticle

(Non-Symmetric) Yetter–Drinfel’d Module Category and Invariant Coinvariant Jacobians

by Zhongwei Wang and Yong Wang

Symmetry 2024, 16(5), 515; https://doi.org/10.3390/sym16050515 - 24 Apr 2024

Viewed by 855

Abstract

In this paper, we generalize the homomorphisms of modules over groups and Lie algebras as being morphisms in the category of (non-symmetric) Yetter–Drinfel’d modules. These module homomorphisms play a key role in the conjecture of Yau. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry Study in Hopf-Type Algebras and Groups)

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