Classification of Non-Degenerate Involutive Set-Theoretic Solutions to the Yang-Baxter Equation with Multipermutation Level Two

2021 ◽  
Author(s):  
Wolfgang Rump
Keyword(s):  
Baxter Equation

scholarly journals Some remarks on invariant Poisson quasi-Nijenhuis structures on Lie groups

2019 ◽  
Vol 16 (07) ◽  
pp. 1950097
Author(s):  
Ghorbanali Haghighatdoost ◽  
Zohreh Ravanpak ◽  
Adel Rezaei-Aghdam
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Baxter Equation ◽  
Complex Structures ◽  
Lie Bialgebra ◽  
Text Structures ◽  
Generalized Complex Structures ◽  
Generalized Complex

We study right-invariant (respectively, left-invariant) Poisson quasi-Nijenhuis structures on a Lie group [Formula: see text] and introduce their infinitesimal counterpart, the so-called r-qn structures on the corresponding Lie algebra [Formula: see text]. We investigate the procedure of the classification of such structures on the Lie algebras and then for clarity of our results we classify, up to a natural equivalence, all [Formula: see text]-[Formula: see text] structures on two types of four-dimensional real Lie algebras. We mention some remarks on the relation between [Formula: see text]-[Formula: see text] structures and the generalized complex structures on the Lie algebras [Formula: see text] and also the solutions of modified Yang–Baxter equation (MYBE) on the double of Lie bialgebra [Formula: see text]. The results are applied to some relevant examples.

scholarly journals Yang-Baxter and the Boost: splitting the difference

2021 ◽  
Vol 11 (3) ◽  
Author(s):  
Marius de Leeuw ◽  
Chiara Paletta ◽  
Anton Pribytok ◽  
Ana L. Retore ◽  
Paul Ryan
Keyword(s):  
Hilbert Space ◽  
Sigma Model ◽  
Spin Chains ◽  
Baxter Equation ◽  
Regular Solutions ◽  
One Dimensional ◽  
Difference Form ◽  
The Difference ◽  
The One

In this paper we continue our classification of regular solutions of the Yang-Baxter equation using the method based on the spin chain boost operator developed in [1]. We provide details on how to find all non-difference form solutions and apply our method to spin chains with local Hilbert space of dimensions two, three and four. We classify all 16\times1616×16 solutions which exhibit \mathfrak{su}(2)\oplus \mathfrak{su}(2)𝔰𝔲(2)⊕𝔰𝔲(2) symmetry, which include the one-dimensional Hubbard model and the SS-matrix of the {AdS}_5 \times {S}^5AdS5×S5 superstring sigma model. In all cases we find interesting novel solutions of the Yang-Baxter equation.

Classification of indecomposable involutive set-theoretic solutions to the Yang–Baxter equation

2020 ◽  
Vol 32 (4) ◽  
pp. 891-903
Author(s):  
Wolfgang Rump
Keyword(s):  
Fundamental Group ◽  
Additive Group ◽  
Baxter Equation ◽  
Adjoint Orbit

AbstractUsing the theory of cycle sets and braces, non-degenerate indecomposable involutive set-theoretic solutions to the Yang–Baxter equation are classified in terms of their universal coverings and their fundamental group. The universal coverings are characterized as braces with an adjoint orbit generating the additive group. Using this description, all coverings of non-degenerate indecomposable cycle sets are classified. The method is illustrated by examples.

scholarly journals Weak Hopf Algebra and Its Quiver Representation

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Muhammad Naseer Khan ◽  
Ahmed Munir ◽  
Muhammad Arshad ◽  
Ahmed Alsanad ◽  
Suheer Al-Hadhrami
Keyword(s):  
Dynamical Systems ◽  
Hopf Algebra ◽  
Canonical Representation ◽  
Weak Hopf Algebra ◽  
Baxter Equation ◽  
Quiver Representation ◽  
Module Theory ◽  
Cayley Digraph

This study induced a weak Hopf algebra from the path coalgebra of a weak Hopf quiver. Moreover, it gave a quiver representation of the said algebra which gives rise to the various structures of the so-called weak Hopf algebra through the quiver. Furthermore, it also showed the canonical representation for each weak Hopf quiver. It was further observed that a Cayley digraph of a Clifford monoid can be embedded in its corresponding weak Hopf quiver of a Clifford monoid. This lead to the development of the foundation structures of weak Hopf algebra. Such quiver representation is useful for the classification of its path coalgebra. Additionally, some structures of module theory of algebra were also given. Such algebras can also be applied for obtaining the solutions of “quantum Yang–Baxter equation” that has many applications in the dynamical systems for finding interesting results.

scholarly journals Asymmetric product of left braces and simplicity; new solutions of the Yang–Baxter equation

2019 ◽  
Vol 21 (08) ◽  
pp. 1850042 ◽  
Author(s):  
D. Bachiller ◽  
F. Cedó ◽  
E. Jespers ◽  
J. Okniński
Keyword(s):  
Multiplicative Group ◽  
Baxter Equation ◽  
Derived Length

The problem of constructing all the non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation (YBE) recently has been reduced to the problem of describing all the left braces. In particular, the classification of all finite left braces is fundamental in order to describe all finite such solutions of the YBE. In this paper, we continue the study of finite simple left braces with the emphasis on the application of the asymmetric product of left braces in order to construct new classes of simple left braces. We do not only construct new classes but also we interpret all previously known constructions as asymmetric products. Moreover, a construction is given of finite simple left braces with a multiplicative group that is solvable of arbitrary derived length.

scholarly journals Parameter-dependent associative Yang–Baxter equations and Poisson brackets

2014 ◽  
Vol 11 (09) ◽  
pp. 1460036 ◽  
Author(s):  
Alexander Odesskii ◽  
Vladimir Rubtsov ◽  
Vladimir Sokolov
Keyword(s):  
General Class ◽  
Baxter Equation ◽  
Poisson Brackets ◽  
Poisson Structures ◽  
One Dimensional ◽  
All Solutions ◽  
Parameter Dependent

We discuss associative analogues of classical Yang–Baxter equation (CYBE) meromorphically dependent on parameters. We discover that such equations enter in a description of a general class of parameter-dependent Poisson structures and double Lie and Poisson structures in sense of Van den Bergh. We propose a classification of all solutions for one-dimensional associative Yang–Baxter equations (AYBE).

scholarly journals Braiding quantum gates from partition algebras

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 311
Author(s):  
Pramod Padmanabhan ◽  
Fumihiko Sugino ◽  
Diego Trancanelli
Keyword(s):  
Statistical Mechanics ◽  
Braid Group ◽  
Quantum Gates ◽  
Entangled States ◽  
Baxter Equation ◽  
Classical Communication ◽  
Qubit System ◽  
Partition Algebras

Unitary braiding operators can be used as robust entangling quantum gates. We introduce a solution-generating technique to solve the (d,m,l)-generalized Yang-Baxter equation, for m/2≤l≤m, which allows to systematically construct such braiding operators. This is achieved by using partition algebras, a generalization of the Temperley-Lieb algebra encountered in statistical mechanics. We obtain families of unitary and non-unitary braiding operators that generate the full braid group. Explicit examples are given for a 2-, 3-, and 4-qubit system, including the classification of the entangled states generated by these operators based on Stochastic Local Operations and Classical Communication.

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