nLab quantum Yang-Baxter equation

For more see at Yang-Baxter equation.

Idea

Given a monoidal category $C$ with tensor product $\otimes$, and an object $V$ in $C$, a quantum Yang-Baxter operator is a morphism of the form

$$R \,\colon\, V \otimes V \longrightarrow V\otimes V$$

which satisfies the following quantum Yang-Baxter equation in $V\otimes V\otimes V$

$$\check R_{12} \, \check R_{13} \, \check R_{23} \;=\; \check R_{23} \, \check R_{13} \, \check R_{12} \,,$$

where the subscripts indicate which tensor factors are being utilized, for instance $R_{12} = R\otimes id_V\in V\otimes V\otimes V$.

This equation is in particular satisfied by the component $\mathcal{R}_{V V}$ at $V$ of any braiding $\mathcal{R}$ on $C$.

Typical categories where the equation is considered are

1. the category of vector spaces when the solutions are called $R$-matrices (or quantum Yang-Baxter matrices),

2. categories of representations of quantum groups; often (a completion of) a quantum group $G$ itself has a particular element $\mathcal{R}$, called a universal $\mathcal{R}$-element, satisfying axioms (quasitriangularity) ensuring that its image in every representation satisfies qYBE

3. the category of sets where one speaks about set theoretic solutions of Yang-Baxter equation.

Remark

(Historical motivation)

The quantum Yang-Baxter equation has been proposed by Baxter in the context of a particular model of statistical mechanics (the 8-vertex model) and called star-triangle relation [Baxter (1978), Baxter (1982)].

Later it has been generalized and axiomatized to a number of contexts: it is most notably satisfied by the universal R-element in a quasitriangular Hopf algebra. In some context it is equivalent to a braid relation for certain transposed matrix. Some solutions to quantum Yang-Baxter equation have good limits in classical mechanics which are classical r-matrices, and the latter satisfy the classical Yang-Baxter equation.

Equation with spectral parameter

With multiplicative spectral parameter, the equation reads

$$\check R_{12} (u) \, \check R_{13} (u v) \, \check R_{23} (v) \;=\; \check R_{23}(v) \, \check R_{13}(u v) \, \check R_{12}(u)$$

where the subscripts indicate which tensor factors are being utilized.

Related concepts

Yang-Baxter equations

• classical Yang-Baxter equation

• quantum Yang-Baxter equation

• dynamical quantum Yang-Baxter equation

• associative Yang-Baxter equation

• infinitesimal braid relation

• set theoretic Yang-Baxter equation

• quasitriangular bialgebra

References

General

The (quantum) Yang-Baxter equation was named (cf. Perk & Au-Yang 2006) by Ludwig Fadeev in the late 1970s, in honor of:

• Chen Ning Yang: Some Exact Results for the Many-Body Problem in one Dimension with Repulsive Delta-Function Interaction, Phys. Rev. Lett. 19 (1967) 1312 [doi:10.1103/PhysRevLett.19.1312]

• Chen Ning Yang: S Matrix for the One-Dimensional $N$-Body Problem with Repulsive or Attractive $\delta$-Function Interaction, Phys. Rev. 168 (1968) 1920 [doi:10.1103/PhysRev.168.1920]

and

• Rodney J. Baxter: Partition function of the Eight-Vertex lattice model, Annals of Physics 70 1 (1972) 193-228 [doi:10.1016/0003-4916(72)90335-1]

• Rodney J. Baxter, Solvable eight-vertex model on an arbitrary planar lattice, Philos. Trans. Royal Society A 289 1359 (1978) [doi:10.1098/rsta.1978.0062]

• Rodney J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press (1982) [webpage, pdf]

Introduction and review:

• Michio Jimbo, Introduction to the Yang-Baxter Equation, Int. J. Modern Physics A 4 15 (1989) 3759-3777 [doi:10.1142/S0217751X89001503]

  reprinted in: Braid Group, Knot Theory and Statistical Mechanics, Advanced Series in Mathematical Physics 9, World Scientific (1991) [doi:10.1142/0796]

• Larry A. Lambe, David E. Radford: Introduction to the quantum Yang-Baxter equation and quantum groups: an algebraic approach, Mathematics and Its Applications 423, Springer, Kluwer (1997) [doi:10.1007/978-1-4615-4109-7]

• Jacques H. H. Perk, Helen Au-Yang: Yang-Baxter Equations, Encyclopedia of Mathematical Physics 5 (2006) 465-473 [arXiv:math-ph/0606053]

Review in the context of braid group representations:

• Camilo Arias Abad, §5.1 Introduction to representations of braid groups, Rev. colomb. mat. vol.49 no.1 (2015) [arXiv:1404.0724, doi:10.15446/recolma.v49n1.54160]

See also:

• Wikipedia, Yang-Baxter equation

• eom, Yang-Baxter equation

Further discussion in the context of quantum groups:

• A. U. Klymik, K. Schmuedgen: Quantum groups and their representations, Springer (1997)

• V. Chari, A. Pressley, A guide to quantum groups, Cambridge Univ. Press (1994)

• V. G. Drinfel'd, Quantum groups, Proceedings of the International Congress of Mathematicians 1986, Vol. 1, 798-820, AMS (1987) [djvu, pdf]

• D. Gurevich, V. Rubtsov: Yang-Baxter equation and deformation of associative and Lie algebras, in: Quantum Groups, Lecture Notes in Math. 1510, Springer (1992) 47-55 [doi:10.1007/BFb0101177]

• P. P. Kulish, Nicolai Reshetikhin, E. K. Sklyanin: Yang-Baxter equation and representation theory: I, Lett. Math. Phys. 5 5 (1981) 393-403 [doi:10.1007/BF02285311]

In the context of topological quantum computing:

• Louis H. Kauffman, Samuel J. Lomonaco, Braiding Operators are Universal Quantum Gates, New Journal of Physics 6 (2004) [doi:10.1088/1367-2630/6/1/134, arXiv:quant-ph/0401090]

• Yong Zhang, Louis H. Kauffman, Mo-Lin Ge: Yang–Baxterizations, Universal Quantum Gates and Hamiltonians, Quantum Inf Process 4 (2005) 159–197 [doi:10.1007/s11128-005-7655-7, arXiv:quant-ph/0502015]

• Dmitry Melnikov, Andrei Mironov, Sergey Mironov, Alexei Morozov, Andrey Morozov: Towards topological quantum computer, Nucl. Phys. B926 (2018) 491-508 [doi:10.1016/j.nuclphysb.2017.11.016, arXiv:1703.00431]

• Nikita Kolganov, Andrey Morozov: Quantum $\mathcal{R}$-matrices as universal qubit gates, Pis’ma v ZhETF, 111, N9 (2020) [doi:10.1134/S0021364020090027, arXiv:2004.07764]

Solutions

Complete list of solutions for the constant quantum Yang-Baxter equation in $dim=2$ (96, falling in 23 classes):

• Jarmo Hietarinta: All solutions to the constant quantum Yang-Baxter equation in two dimensions, Physics Letters A 165 3 (1992) 245-251 [doi;10.1016/0375-9601(92)90044-M]

• Jarmo Hietarinta: The complete solution to the constant quantum Yang-Baxter equation in two dimensions, talk at 19th International Colloquium on Group-theoretical Methods in Physics (1992) [arXiv:hep-th/9210067]

Discussion of certain quantum R-matrices as universal quantum gates for topological quantum computing (where one is interested in unitary solutions):

• Louis H. Kauffman, Samuel J. Lomonaco: Braiding Operators are Universal Quantum Gates, New Journal of Physics 6 (2004) [doi:10.1088/1367-2630/6/1/134, arXiv:quant-ph/0401090]

Classification of all unitary solutions in $dim=4$:

• H. A. Dye: Unitary Solutions to the Yang-Baxter Equation in Dimension Four, Quantum Information Processing 2 (2003) (117-152) [arXiv:quant-ph/0211050, doi:10.1023/A:1025843426102]

Discussion of all involutive solutions, yielding representations of a symmetric group:

• Gandalf Lechner, Ulrich Pennig, Simon Wood: Yang-Baxter representations of the infinite symmetric group, Adv. Math. 355 (2019) 106769 [arXiv:1707.00196, doi:10.1016/j.aim.2019.106769]

  p 15: “Yang-Baxter representations are reducible, but decomposing them gives representations which are no longer of Yang-Baxter form. Conversely, taking direct sums of Yang-Baxter representations is not compatible with the Yang-Baxter equation either.”

• Gandalf Lechner: All involutive solutions of the Yang-Baxter equation, talk at Advances in Mathematics and Theoretical Physics (2017) [pdf, pdf]