Schoof–Elkies–Atkin algorithm: Difference between revisions
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In [[physics]], the '''Yang–Baxter equation''' (or '''star-triangle relation''') is a consistency equation which was first introduced in the field of [[statistical mechanics]]. It's depends on the idea that in some scattering situations, particles may preserve their momentum in price of changing their quantum internal states. It states that a matrix <math> R</math>, acting on two out of three objects, satisfies | |||
:<math>(R\otimes \mathbf{1})(\mathbf{1}\otimes R)(R\otimes \mathbf{1}) =(\mathbf{1}\otimes R)(R\otimes \mathbf{1})(\mathbf{1}\otimes R)</math> | |||
In one dimensional quantum systems, <math>R</math> is the scattering matrix and if it satisfies the Yang-Baxter equation then the system is [[Integrable system#Quantum integrable systems|integrable]]. The Yang-Baxter equation also shows up when discussing [[knot theory]] and the [[braid groups]] where <math> R</math> corresponds to swapping two strands. Since one can swap three strands two different ways, the Yang-Baxter equation enforces that both paths are the same. | |||
[[File:Illustration of Yang Baxter Equation.png|thumb|Illustration of Yang Baxter Equation]] | |||
It takes its name from independent work of [[C. N. Yang]] from 1968, and [[R. J. Baxter]] from 1971. | |||
==Parameter-dependent Yang–Baxter equation== | |||
Let <math>A</math> be a [[unital algebra|unital]] [[associative]] [[Algebra over a field|algebra]]. The parameter-dependent Yang–Baxter equation is an equation for <math>R(u)</math>, a parameter-dependent [[inverse element|invertible]] element of the [[tensor product of algebras|tensor product]] <math>A \otimes A</math> (here, <math>u</math> is the parameter, which usually ranges over all real numbers in the case of an additive parameter, or over all positive real numbers in the case of a multiplicative parameter). The Yang–Baxter equation is | |||
:<math>R_{12}(u) \ R_{13}(u+v) \ R_{23}(v) = R_{23}(v) \ R_{13}(u+v) \ R_{12}(u),</math> | |||
for all values of <math>u</math> and <math>v</math>, in the case of an additive parameter. At some value of the parameter <math>R(u)</math> can turn into one dimensional projector, this gives rise to quantum determinant. For multiplicative parameter Yang–Baxter equation is | |||
:<math>R_{12}(u) \ R_{13}(uv) \ R_{23}(v) = R_{23}(v) \ R_{13}(uv) \ R_{12}(u),</math> | |||
for all values of <math>u</math> and <math>v</math>, where <math>R_{12}(w) = \phi_{12}(R(w))</math>, <math>R_{13}(w) = \phi_{13}(R(w))</math>, and <math>R_{23}(w) = \phi_{23}(R(w))</math>, for all values of the parameter <math>w</math>, and <math>\phi_{12} : A \otimes A \to A \otimes A \otimes A</math>, <math>\phi_{13} : A \otimes A \to A \otimes A \otimes A</math>, and <math>\phi_{23} : A \otimes A \to A \otimes A \otimes A</math> are algebra morphisms determined by | |||
:<math>\phi_{12}(a \otimes b) = a \otimes b \otimes 1,</math> | |||
:<math>\phi_{13}(a \otimes b) = a \otimes 1 \otimes b,</math> | |||
:<math>\phi_{23}(a \otimes b) = 1 \otimes a \otimes b.</math> | |||
In some cases the determinant of <math>R (u) </math> can vanish at specific values of the spectral parameter <math> u=u_{0} </math>. Some <math>R </math> matrices turn into one dimensional projector at | |||
<math> u=u_{0} </math>. In this case quantum determinant can be defined. | |||
==Parameter-independent Yang–Baxter equation== | |||
Let <math>A</math> be a unital associative algebra. The parameter-independent Yang–Baxter equation is an equation for <math>R</math>, an invertible element of the tensor product <math>A \otimes A</math>. The Yang–Baxter equation is | |||
:<math>R_{12} \ R_{13} \ R_{23} = R_{23} \ R_{13} \ R_{12},</math> | |||
where <math>R_{12} = \phi_{12}(R)</math>, <math>R_{13} = \phi_{13}(R)</math>, and <math>R_{23} = \phi_{23}(R)</math>. | |||
Let <math>V</math> be a [[module (mathematics)|module]] of <math>A</math>. Let <math>T : V \otimes V \to V \otimes V</math> be the linear map satisfying <math>T(x \otimes y) = y \otimes x</math> for all <math>x, y \in V</math>. Then a [[group representation|representation]] of the [[braid group]], <math>B_n</math>, can be constructed on <math>V^{\otimes n}</math> by <math>\sigma_i = 1^{\otimes i-1} \otimes \check{R} \otimes 1^{\otimes n-i-1}</math> for <math>i = 1,\dots,n-1</math>, where <math>\check{R} = T \circ R</math> on <math>V \otimes V</math>. This representation can be used to determine quasi-invariants of [[braid theory|braids]], [[knot (mathematics)|knots]] and [[link (knot theory)|links]]. | |||
==See also== | |||
* [[Lie bialgebra]] | |||
* [[Yangian]] | |||
==References== | |||
* H.-D. Doebner, J.-D. Hennig, eds, ''Quantum groups, Proceedings of the 8th International Workshop on Mathematical Physics, Arnold Sommerfeld Institute, Clausthal, FRG, 1989'', Springer-Verlag Berlin, ISBN 3-540-53503-9. | |||
* Vyjayanthi Chari and Andrew Pressley, ''A Guide to Quantum Groups'', (1994), Cambridge University Press, Cambridge ISBN 0-521-55884-0. | |||
* Jacques H.H. Perk and Helen Au-Yang, "Yang–Baxter Equations", (2006), {{arxiv|math-ph/0606053}}. | |||
==External links== | |||
* {{springer|title=Yang-Baxter equation|id=p/y099020}} | |||
{{DEFAULTSORT:Yang-Baxter equation}} | |||
[[Category:Monoidal categories]] | |||
[[Category:Statistical mechanics]] | |||
[[Category:Equations of physics]] | |||
[[Category:Exactly solvable models]] | |||
[[Category:Conformal field theory]] |
Latest revision as of 17:47, 20 April 2013
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In physics, the Yang–Baxter equation (or star-triangle relation) is a consistency equation which was first introduced in the field of statistical mechanics. It's depends on the idea that in some scattering situations, particles may preserve their momentum in price of changing their quantum internal states. It states that a matrix , acting on two out of three objects, satisfies
In one dimensional quantum systems, is the scattering matrix and if it satisfies the Yang-Baxter equation then the system is integrable. The Yang-Baxter equation also shows up when discussing knot theory and the braid groups where corresponds to swapping two strands. Since one can swap three strands two different ways, the Yang-Baxter equation enforces that both paths are the same.
It takes its name from independent work of C. N. Yang from 1968, and R. J. Baxter from 1971.
Parameter-dependent Yang–Baxter equation
Let be a unital associative algebra. The parameter-dependent Yang–Baxter equation is an equation for , a parameter-dependent invertible element of the tensor product (here, is the parameter, which usually ranges over all real numbers in the case of an additive parameter, or over all positive real numbers in the case of a multiplicative parameter). The Yang–Baxter equation is
for all values of and , in the case of an additive parameter. At some value of the parameter can turn into one dimensional projector, this gives rise to quantum determinant. For multiplicative parameter Yang–Baxter equation is
for all values of and , where , , and , for all values of the parameter , and , , and are algebra morphisms determined by
In some cases the determinant of can vanish at specific values of the spectral parameter . Some matrices turn into one dimensional projector at . In this case quantum determinant can be defined.
Parameter-independent Yang–Baxter equation
Let be a unital associative algebra. The parameter-independent Yang–Baxter equation is an equation for , an invertible element of the tensor product . The Yang–Baxter equation is
Let be a module of . Let be the linear map satisfying for all . Then a representation of the braid group, , can be constructed on by for , where on . This representation can be used to determine quasi-invariants of braids, knots and links.
See also
References
- H.-D. Doebner, J.-D. Hennig, eds, Quantum groups, Proceedings of the 8th International Workshop on Mathematical Physics, Arnold Sommerfeld Institute, Clausthal, FRG, 1989, Springer-Verlag Berlin, ISBN 3-540-53503-9.
- Vyjayanthi Chari and Andrew Pressley, A Guide to Quantum Groups, (1994), Cambridge University Press, Cambridge ISBN 0-521-55884-0.
- Jacques H.H. Perk and Helen Au-Yang, "Yang–Baxter Equations", (2006), Template:Arxiv.
External links
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