Self-distributive structures, braces & the Yang-Baxter equation
- URL: http://arxiv.org/abs/2409.20479v1
- Date: Mon, 30 Sep 2024 16:40:41 GMT
- Title: Self-distributive structures, braces & the Yang-Baxter equation
- Authors: Anastasia Doikou,
- Abstract summary: The theory of the set-theoretic Yang-Baxter equation is reviewed from a purely algebraic point of view.
The notion of braces is also presented as the suitable algebraic structure associated to involutive set-theoretic solutions.
The quantum algebra as well as the integrability of Baxterized involutive set-theoretic solutions is also discussed.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: The theory of the set-theoretic Yang-Baxter equation is reviewed from a purely algebraic point of view. We recall certain algebraic structures called shelves, racks and quandles. These objects satisfy a self-distributivity condition and lead to solutions of the Yang-Baxter equation. We also recall that non-involutive solutions of the braid equation are obtained from shelf and rack solutions by a suitable parametric twist, whereas all involutive set-theoretic solutions are reduced to the flip map via a parametric twist. The notion of braces is also presented as the suitable algebraic structure associated to involutive set-theoretic solutions. The quantum algebra as well as the integrability of Baxterized involutive set-theoretic solutions is also discussed. The explicit form of the Drinfel'd twist is presented allowing the derivation of general set-theoretic solutions.
Related papers
- Solving the Yang-Baxter, tetrahedron and higher simplex equations using Clifford algebras [0.0]
Yang-Baxter equation, Zamalodchikov's tetrahedron equation and Bazhanov-Stroganov equation are special cases.
We describe a universal method to solve these equations using Clifford algebras.
arXiv Detail & Related papers (2024-04-17T15:56:16Z) - Holomorphic Floer theory I: exponential integrals in finite and infinite dimensions [0.0]
We discuss exponential integrals and related wall-crossing structures.
We develop the corresponding theories in particular generalizing Morse-Novikov theory to the holomorphic case.
As a corollary, perturbative expansions of exponential integrals are resurgent.
arXiv Detail & Related papers (2024-02-12T00:21:31Z) - Radiative transport in a periodic structure with band crossings [47.82887393172228]
We derive the semi-classical model for the Schr"odinger equation in arbitrary spatial dimensions.
We consider both deterministic and random scenarios.
As a specific application, we deduce the effective dynamics of a wave packet in graphene with randomness.
arXiv Detail & Related papers (2024-02-09T23:34:32Z) - A Pedestrian's Way to Baxter's Bethe Ansatz for the Periodic XYZ Chain [2.1797546092115803]
We construct a set of chiral vectors with fixed number of kinks.
Under roots of unity conditions, the Hilbert space has an invariant subspace.
We propose a Bethe ansatz based on the action of the Hamiltonian on the chiral vectors.
arXiv Detail & Related papers (2023-11-30T19:48:23Z) - An algebraic formulation of nonassociative quantum mechanics [0.0]
We develop a version of quantum mechanics that can handle nonassociative algebras of observables.
Our approach is naturally probabilistic and is based on using the universal enveloping algebra of a general nonassociative algebra.
arXiv Detail & Related papers (2023-11-07T01:36:23Z) - Enriching Diagrams with Algebraic Operations [49.1574468325115]
We extend diagrammatic reasoning in monoidal categories with algebraic operations and equations.
We show how this construction can be used for diagrammatic reasoning of noise in quantum systems.
arXiv Detail & Related papers (2023-10-17T14:12:39Z) - Algebras of actions in an agent's representations of the world [46.74201905814679]
We use our framework to reproduce the symmetry-based representations from the symmetry-based disentangled representation learning formalism.<n>We then study the algebras of the transformations of worlds with features that occur in simple reinforcement learning scenarios.<n>Using computational methods, that we developed, we extract the algebras of the transformations of these worlds and classify them according to their properties.
arXiv Detail & Related papers (2023-10-02T18:24:51Z) - Bethe ansatz solutions and hidden $sl(2)$ algebraic structure for a
class of quasi-exactly solvable systems [0.638421840998693]
We revisit a class of models for which the odd solutions were largely missed previously in the literature.
We present a systematic and unified treatment for the odd and even sectors of these models.
We also make progress in the analysis of solutions to the Bethe ansatz equations in the spaces of model parameters.
arXiv Detail & Related papers (2023-09-21T02:04:44Z) - Comparison of Single- and Multi- Objective Optimization Quality for
Evolutionary Equation Discovery [77.34726150561087]
Evolutionary differential equation discovery proved to be a tool to obtain equations with less a priori assumptions.
The proposed comparison approach is shown on classical model examples -- Burgers equation, wave equation, and Korteweg - de Vries equation.
arXiv Detail & Related papers (2023-06-29T15:37:19Z) - Universal Braiding Quantum Gates [0.0]
Unitary solutions of the Yang-Baxter equation are of particular interest as quantum gates for topological quantum computers.
We classify a family of solutions to certain generalized Yang-Baxter equations and prove that certain instances of the equation only have solutions that are scalar multiples of the identity.
arXiv Detail & Related papers (2023-04-03T04:03:23Z) - Calculation of the wave functions of a quantum asymmetric top using the
noncommutative integration method [0.0]
We obtain a complete set of solutions to the Schrodinger equation for a quantum asymmetric top in Euler angles.
The spectrum of an asymmetric top is obtained from the condition that the solutions are in with respect to a special irreducible $lambda$-representation of the rotation group.
arXiv Detail & Related papers (2022-11-27T12:38:22Z) - Harmonic oscillator coherent states from the orbit theory standpoint [0.0]
We show that analogs of coherent states constructed by the noncommutative integration can be expressed in terms of the solution of a system of differential equations on the Lie group.
The solutions constructed are directly related to irreducible representation of the Lie algebra on the Hilbert space functions on the Lagrangian submanifold to the orbit of the coadjoint representation.
arXiv Detail & Related papers (2022-11-20T17:15:02Z) - A Complete Equational Theory for Quantum Circuits [58.720142291102135]
We introduce the first complete equational theory for quantum circuits.
Two circuits represent the same unitary map if and only if they can be transformed one into the other using the equations.
arXiv Detail & Related papers (2022-06-21T17:56:31Z) - Qudit lattice surgery [91.3755431537592]
We observe that lattice surgery, a model of fault-tolerant qubit computation, generalises straightforwardly to arbitrary finite-dimensional qudits.
We relate the model to the ZX-calculus, a diagrammatic language based on Hopf-Frobenius algebras.
arXiv Detail & Related papers (2022-04-27T23:41:04Z) - Negative Translations of Orthomodular Lattices and Their Logic [0.0]
We introduce residuated ortholattices as a generalization of -- and environment for the investigation of -- orthomodular lattices.
We show that residuated ortholattices are the equivalent algebraic semantics of an algebraizable propositional logic.
arXiv Detail & Related papers (2021-06-07T14:35:27Z) - Bernstein-Greene-Kruskal approach for the quantum Vlasov equation [91.3755431537592]
The one-dimensional stationary quantum Vlasov equation is analyzed using the energy as one of the dynamical variables.
In the semiclassical case where quantum tunneling effects are small, an infinite series solution is developed.
arXiv Detail & Related papers (2021-02-18T20:55:04Z) - Sub-bosonic (deformed) ladder operators [62.997667081978825]
We present a class of deformed creation and annihilation operators that originates from a rigorous notion of fuzziness.
This leads to deformed, sub-bosonic commutation relations inducing a simple algebraic structure with modified eigenenergies and Fock states.
In addition, we investigate possible consequences of the introduced formalism in quantum field theories, as for instance, deviations from linearity in the dispersion relation for free quasibosons.
arXiv Detail & Related papers (2020-09-10T20:53:58Z) - Competitive Mirror Descent [67.31015611281225]
Constrained competitive optimization involves multiple agents trying to minimize conflicting objectives, subject to constraints.
We propose competitive mirror descent (CMD): a general method for solving such problems based on first order information.
As a special case we obtain a novel competitive multiplicative weights algorithm for problems on the positive cone.
arXiv Detail & Related papers (2020-06-17T22:11:35Z) - Operator-algebraic renormalization and wavelets [62.997667081978825]
We construct the continuum free field as the scaling limit of Hamiltonian lattice systems using wavelet theory.
A renormalization group step is determined by the scaling equation identifying lattice observables with the continuum field smeared by compactly supported wavelets.
arXiv Detail & Related papers (2020-02-04T18:04:51Z) - A refinement of Reznick's Positivstellensatz with applications to
quantum information theory [72.8349503901712]
In Hilbert's 17th problem Artin showed that any positive definite in several variables can be written as the quotient of two sums of squares.
Reznick showed that the denominator in Artin's result can always be chosen as an $N$-th power of the squared norm of the variables.
arXiv Detail & Related papers (2019-09-04T11:46:26Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.