R matrix

In statistical physics , matrices that follow the Yang-Baxter equation (after CN Yang and Rodney Baxter ) are: ${\displaystyle R\in Mat(n)}$

${\displaystyle R^{12}R^{13}R^{23}=R^{23}R^{13}R^{12}}$

suffice, called R-matrices .

In mathematics , R-matrices are used to construct quantum invariants in knot theory .

Description of the Yang-Baxter equation in coordinates

Illustration of the Yang-Baxter equation

A matrix with entries can be understood as the endomorphism of the with base , i.e. ${\displaystyle n^{2}\times n^{2}}$ ${\displaystyle R}$ ${\displaystyle r_{ij}^{kl}}$ ${\displaystyle \mathbb {C} ^{n}\otimes \mathbb {C} ^{n}}$ ${\displaystyle e_{i}\otimes e_{j}}$

${\displaystyle R(e_{i}\otimes e_{j})=\sum _{k,l}r_{ij}^{kl}e_{k}\otimes e_{l}}$.

The Yang-Baxter equation can be written as

${\displaystyle R^{12}R^{13}R^{23}=R^{23}R^{13}R^{12}}$,

where is the endomorphism of acting on the factors as and on the third factor as identity mapping . So ${\displaystyle R^{ij}}$ ${\displaystyle \mathbb {C} ^{n}\otimes \mathbb {C} ^{n}\otimes \mathbb {C} ^{n}}$ ${\displaystyle i,j}$ ${\displaystyle R}$

${\displaystyle R^{12}=R\otimes id,R^{23}=id\otimes R}$

and

${\displaystyle R^{13}(e_{i}\otimes e_{j}\otimes e_{k})=\sum _{a,b}r_{ik}^{ab}e_{a}\otimes e_{j}\otimes e_{b}}$.

R-matrices in quantum mechanics

A one-dimensional quantum mechanical system is integrable if and only if its scatter matrix satisfies the Yang-Baxter equation, i.e. if it is an R matrix.

R-matrices in knot theory

Any R-matrix can be used to construct a quantum invariant of nodes.

literature

• Yang-Baxter equation . In: Michiel Hazewinkel (Ed.): Encyclopedia of Mathematics. Springer, 2001, ISBN 978-1-55608-010-4 .
• J. Park, H. Au-Yang: Yang-Baxter equations. In: J.-P. Françoise, GL Naber, Tsou ST (Eds.): Encyclopedia of Mathematical Physics. Volume 5, Elsevier, Oxford 2006, ISBN 978-0-12-512666-3 , pp. 465-473.
• M. Jimbo: Quantum R matrix for the generalized Toda system. In: Comm. Math. Phys. 102, No. 4, 1986, pp. 537-547, doi : 10.1007 / BF01221646 .

Individual evidence

1. ↑ Yang, Some exact results for the many-body problem in one dimension with delta-function interaction, Phys. Rev. Lett., Volume 19, 1967, pp. 1312-1314, doi : 10.1103 / PhysRevLett.19.1312
2. ↑ Baxter, Solvable eight-vertex model on an arbitrary planar lattice, Phil. Trans. Royal Soc., Vol. 289, 1978, pp. 315-346, doi : 10.1098 / rsta.1978.0062 , JSTOR 75051