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Yang–Baxter equation
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<h2 id="history">History</h2> <p>According to <a href="/facts/Michio_Jimbo/Ij6q4FoL">Michio Jimbo</a>,<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> the Yang–Baxter equation (YBE) manifested itself in the works of J. B. McGuire<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> in 1964 and <a href="/facts/C._N._Yang/VjoM4raz">C. N. Yang</a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> in 1967. They considered a quantum mechanical many-body problem on a line having c ∑ i < j δ ( x i − x j ) {\displaystyle c\sum _{i<j}\delta (x_{i}-x_{j})} as the potential. Using the <a href="/facts/Bethe_ansatz/wSEGLbK6">Bethe ansatz</a> techniques, they found that the scattering matrix factorized to that of the two-body problem, and determined it exactly. Here YBE arises as the consistency condition for the factorization. </p><p>In <a href="/facts/Statistical_mechanics/6zMsNYYG">statistical mechanics</a>, the source of YBE probably goes back to Onsager's star-triangle relation, briefly mentioned in the introduction to his solution of the Ising model<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> in 1944. The hunt for solvable lattice models has been actively pursued since then, culminating in <a href="/facts/Rodney_Baxter/f4CASSbq">Rodney Baxter</a>'s solution of the eight vertex model<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> in 1972. </p><p>Another line of development was the theory of factorized <i>S</i>-matrix in two dimensional quantum field theory.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> Alexander B. Zamolodchikov pointed out<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> that the algebraic mechanics working here is the same as that in the Baxter's and others' works. </p><p>The YBE has also manifested itself in a study of <a href="/facts/Young_symmetrizer/S6yrvHvq">Young operators</a> in the <a href="/facts/Group_ring/TJlkt5Te">group algebra</a> C [ S n ] {\displaystyle \mathbb {C} [S_{n}]} of the <a href="/facts/Symmetric_group/RUksK5l5">symmetric group</a> in the work of <a href="/facts/Algimantas_Adolfas_Jucys/t0PdMebW"> A. A. Jucys</a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> in 1966. </p> <h2 id="general-form-of-the-parameter-dependent-yangbaxter-equation">General form of the parameter-dependent Yang–Baxter equation</h2> <p>Let A {\displaystyle A} be a <a href="/facts/Unital_algebra/8T8ulWJb">unital</a> <a href="/facts/Associative_algebra/DRfoF6KV">associative algebra</a>. In its most general form, the parameter-dependent Yang–Baxter equation is an equation for R ( u , u ′ ) {\displaystyle R(u,u')} , a parameter-dependent element of the <a href="/facts/Tensor_product_of_algebras/TdWszpzf">tensor product</a> A ⊗ A {\displaystyle A\otimes A} (here, u {\displaystyle u} and u ′ {\displaystyle u'} are the parameters, which usually range over the <a href="/facts/Real_number/R02gw5Pb">real numbers</a> ℝ in the case of an additive parameter, or over <a href="/facts/Positive_real_numbers/VFXU56H9">positive real numbers</a> ℝ+ in the case of a multiplicative parameter). </p><p>Let R i j ( u , u ′ ) = ϕ i j ( R ( u , u ′ ) ) {\displaystyle R_{ij}(u,u')=\phi _{ij}(R(u,u'))} for 1 ≤ i < j ≤ 3 {\displaystyle 1\leq i<j\leq 3} , with algebra <a href="/facts/Homomorphism/0gyqdEse">homomorphisms</a> ϕ i j : A ⊗ A → A ⊗ A ⊗ A {\displaystyle \phi _{ij}:A\otimes A\to A\otimes A\otimes A} determined by </p> ϕ 12 ( a ⊗ b ) = a ⊗ b ⊗ 1 , {\displaystyle \phi _{12}(a\otimes b)=a\otimes b\otimes 1,} ϕ 13 ( a ⊗ b ) = a ⊗ 1 ⊗ b , {\displaystyle \phi _{13}(a\otimes b)=a\otimes 1\otimes b,} ϕ 23 ( a ⊗ b ) = 1 ⊗ a ⊗ b . {\displaystyle \phi _{23}(a\otimes b)=1\otimes a\otimes b.} <p>The general form of the Yang–Baxter equation is </p> R 12 ( u 1 , u 2 ) R 13 ( u 1 , u 3 ) R 23 ( u 2 , u 3 ) = R 23 ( u 2 , u 3 ) R 13 ( u 1 , u 3 ) R 12 ( u 1 , u 2 ) , {\displaystyle R_{12}(u_{1},u_{2})\ R_{13}(u_{1},u_{3})\ R_{23}(u_{2},u_{3})=R_{23}(u_{2},u_{3})\ R_{13}(u_{1},u_{3})\ R_{12}(u_{1},u_{2}),} <p>for all values of u 1 {\displaystyle u_{1}} , u 2 {\displaystyle u_{2}} and u 3 {\displaystyle u_{3}} . </p> <h3>Parameter-independent form</h3> <p>Let A {\displaystyle A} be a unital associative algebra. The parameter-independent Yang–Baxter equation is an equation for R {\displaystyle R} , an invertible element of the tensor product A ⊗ A {\displaystyle A\otimes A} . The Yang–Baxter equation is </p> R 12 R 13 R 23 = R 23 R 13 R 12 , {\displaystyle R_{12}\ R_{13}\ R_{23}=R_{23}\ R_{13}\ R_{12},} <p>where R 12 = ϕ 12 ( R ) {\displaystyle R_{12}=\phi _{12}(R)} , R 13 = ϕ 13 ( R ) {\displaystyle R_{13}=\phi _{13}(R)} , and R 23 = ϕ 23 ( R ) {\displaystyle R_{23}=\phi _{23}(R)} . </p> <h3>With respect to a basis</h3> <p>Often the unital associative algebra is the <a href="/facts/Endomorphism_algebra/K1UgpnJK">algebra of endomorphisms</a> of a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> V {\displaystyle V} over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> k {\displaystyle k} , that is, A = End ( V ) {\displaystyle A={\text{End}}(V)} . With respect to a <a href="/facts/Basis_(linear_algebra)/89IPoN6c">basis</a> { e i } {\displaystyle \{e_{i}\}} of V {\displaystyle V} , the components of the matrices R ∈ End ( V ) ⊗ End ( V ) ≅ End ( V ⊗ V ) {\displaystyle R\in {\text{End}}(V)\otimes {\text{End}}(V)\cong {\text{End}}(V\otimes V)} are written R i j k l {\displaystyle R_{ij}^{kl}} , which is the component associated to the map e i ⊗ e j ↦ e k ⊗ e l {\displaystyle e_{i}\otimes e_{j}\mapsto e_{k}\otimes e_{l}} . Omitting parameter dependence, the component of the Yang–Baxter equation associated to the map e a ⊗ e b ⊗ e c ↦ e d ⊗ e e ⊗ e f {\displaystyle e_{a}\otimes e_{b}\otimes e_{c}\mapsto e_{d}\otimes e_{e}\otimes e_{f}} reads </p> ( R 12 ) i j d e ( R 13 ) a k i f ( R 23 ) b c j k = ( R 23 ) j k e f ( R 13 ) i c d k ( R 12 ) a b i j . {\displaystyle (R_{12})_{ij}^{de}(R_{13})_{ak}^{if}(R_{23})_{bc}^{jk}=(R_{23})_{jk}^{ef}(R_{13})_{ic}^{dk}(R_{12})_{ab}^{ij}.} <h2 id="alternate-form-and-representations-of-the-braid-group">Alternate form and representations of the braid group</h2> <p>Let V {\displaystyle V} be a <a href="/facts/Module_(mathematics)/jP0LIhFL">module</a> of A {\displaystyle A} , and P i j = ϕ i j ( P ) {\displaystyle P_{ij}=\phi _{ij}(P)} . Let P : V ⊗ V → V ⊗ V {\displaystyle P:V\otimes V\to V\otimes V} be the linear map satisfying P ( x ⊗ y ) = y ⊗ x {\displaystyle P(x\otimes y)=y\otimes x} for all x , y ∈ V {\displaystyle x,y\in V} . The Yang–Baxter equation then has the following alternate form in terms of R ˇ ( u , u ′ ) = P ∘ R ( u , u ′ ) {\displaystyle {\check {R}}(u,u')=P\circ R(u,u')} on V ⊗ V {\displaystyle V\otimes V} . </p> ( 1 ⊗ R ˇ ( u 1 , u 2 ) ) ( R ˇ ( u 1 , u 3 ) ⊗ 1 ) ( 1 ⊗ R ˇ ( u 2 , u 3 ) ) = ( R ˇ ( u 2 , u 3 ) ⊗ 1 ) ( 1 ⊗ R ˇ ( u 1 , u 3 ) ) ( R ˇ ( u 1 , u 2 ) ⊗ 1 ) {\displaystyle (\mathbf {1} \otimes {\check {R}}(u_{1},u_{2}))({\check {R}}(u_{1},u_{3})\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(u_{2},u_{3}))=({\check {R}}(u_{2},u_{3})\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(u_{1},u_{3}))({\check {R}}(u_{1},u_{2})\otimes \mathbf {1} )} . <p>Alternatively, we can express it in the same notation as above, defining R ˇ i j ( u , u ′ ) = ϕ i j ( R ˇ ( u , u ′ ) ) {\displaystyle {\check {R}}_{ij}(u,u')=\phi _{ij}({\check {R}}(u,u'))} , in which case the alternate form is </p> R ˇ 23 ( u 1 , u 2 ) R ˇ 12 ( u 1 , u 3 ) R ˇ 23 ( u 2 , u 3 ) = R ˇ 12 ( u 2 , u 3 ) R ˇ 23 ( u 1 , u 3 ) R ˇ 12 ( u 1 , u 2 ) . {\displaystyle {\check {R}}_{23}(u_{1},u_{2})\ {\check {R}}_{12}(u_{1},u_{3})\ {\check {R}}_{23}(u_{2},u_{3})={\check {R}}_{12}(u_{2},u_{3})\ {\check {R}}_{23}(u_{1},u_{3})\ {\check {R}}_{12}(u_{1},u_{2}).} <p>In the parameter-independent special case where R ˇ {\displaystyle {\check {R}}} does not depend on parameters, the equation reduces to </p> ( 1 ⊗ R ˇ ) ( R ˇ ⊗ 1 ) ( 1 ⊗ R ˇ ) = ( R ˇ ⊗ 1 ) ( 1 ⊗ R ˇ ) ( R ˇ ⊗ 1 ) {\displaystyle (\mathbf {1} \otimes {\check {R}})({\check {R}}\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}})=({\check {R}}\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}})({\check {R}}\otimes \mathbf {1} )} , <p>and (if R {\displaystyle R} is invertible) a <a href="/facts/Group_representation/FsuuDagQ">representation</a> of the <a href="/facts/Braid_group/rn5323Kx">braid group</a>, B n {\displaystyle B_{n}} , can be constructed on V ⊗ n {\displaystyle V^{\otimes n}} by σ i = 1 ⊗ i − 1 ⊗ R ˇ ⊗ 1 ⊗ n − i − 1 {\displaystyle \sigma _{i}=1^{\otimes i-1}\otimes {\check {R}}\otimes 1^{\otimes n-i-1}} for i = 1 , … , n − 1 {\displaystyle i=1,\dots ,n-1} . This representation can be used to determine quasi-invariants of <a href="/facts/Braid_theory/rn5323Kx">braids</a>, <a href="/facts/Knot_(mathematics)/UMtKJ36m">knots</a> and <a href="/facts/Link_(knot_theory)/hoFDYmVj">links</a>. </p> <h2 id="symmetry">Symmetry</h2> <p>Solutions to the Yang–Baxter equation are often constrained by requiring the R {\displaystyle R} matrix to be invariant under the action of a <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> G {\displaystyle G} . For example, in the case G = G L ( V ) {\displaystyle G=GL(V)} and R ( u , u ′ ) ∈ End ( V ⊗ V ) {\displaystyle R(u,u')\in {\text{End}}(V\otimes V)} , the only G {\displaystyle G} -invariant maps in End ( V ⊗ V ) {\displaystyle {\text{End}}(V\otimes V)} are the identity I {\displaystyle I} and the permutation map P {\displaystyle P} . The general form of the R {\displaystyle R} -matrix is then R ( u , u ′ ) = A ( u , u ′ ) I + B ( u , u ′ ) P {\displaystyle R(u,u')=A(u,u')I+B(u,u')P} for scalar functions A , B {\displaystyle A,B} . </p><p>The Yang–Baxter equation is homogeneous in parameter dependence in the sense that if one defines R ′ ( u i , u j ) = f ( u i , u j ) R ( u i , u j ) {\displaystyle R'(u_{i},u_{j})=f(u_{i},u_{j})R(u_{i},u_{j})} , where f {\displaystyle f} is a scalar function, then R ′ {\displaystyle R'} also satisfies the Yang–Baxter equation. </p><p>The argument space itself may have symmetry. For example translation invariance enforces that the dependence on the arguments ( u , u ′ ) {\displaystyle (u,u')} must be dependent only on the translation-invariant difference u − u ′ {\displaystyle u-u'} , while scale invariance enforces that R {\displaystyle R} is a function of the scale-invariant ratio u / u ′ {\displaystyle u/u'} . </p> <h2 id="parametrizations-and-example-solutions">Parametrizations and example solutions</h2> <p>A common ansatz for computing solutions is the difference property, R ( u , u ′ ) = R ( u − u ′ ) {\displaystyle R(u,u')=R(u-u')} , where R depends only on a single (additive) parameter. Equivalently, taking logarithms, we may choose the parametrization R ( u , u ′ ) = R ( u / u ′ ) {\displaystyle R(u,u')=R(u/u')} , in which case R is said to depend on a multiplicative parameter. In those cases, we may reduce the YBE to two free parameters in a form that facilitates computations: </p> R 12 ( u ) R 13 ( u + v ) R 23 ( v ) = R 23 ( v ) R 13 ( u + v ) R 12 ( u ) , {\displaystyle R_{12}(u)\ R_{13}(u+v)\ R_{23}(v)=R_{23}(v)\ R_{13}(u+v)\ R_{12}(u),} <p>for all values of u {\displaystyle u} and v {\displaystyle v} . For a multiplicative parameter, the Yang–Baxter equation is </p> R 12 ( u ) R 13 ( u v ) R 23 ( v ) = R 23 ( v ) R 13 ( u v ) R 12 ( u ) , {\displaystyle R_{12}(u)\ R_{13}(uv)\ R_{23}(v)=R_{23}(v)\ R_{13}(uv)\ R_{12}(u),} <p>for all values of u {\displaystyle u} and v {\displaystyle v} . </p><p>The braided forms read as: </p> ( 1 ⊗ R ˇ ( u ) ) ( R ˇ ( u + v ) ⊗ 1 ) ( 1 ⊗ R ˇ ( v ) ) = ( R ˇ ( v ) ⊗ 1 ) ( 1 ⊗ R ˇ ( u + v ) ) ( R ˇ ( u ) ⊗ 1 ) {\displaystyle (\mathbf {1} \otimes {\check {R}}(u))({\check {R}}(u+v)\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(v))=({\check {R}}(v)\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(u+v))({\check {R}}(u)\otimes \mathbf {1} )} ( 1 ⊗ R ˇ ( u ) ) ( R ˇ ( u v ) ⊗ 1 ) ( 1 ⊗ R ˇ ( v ) ) = ( R ˇ ( v ) ⊗ 1 ) ( 1 ⊗ R ˇ ( u v ) ) ( R ˇ ( u ) ⊗ 1 ) {\displaystyle (\mathbf {1} \otimes {\check {R}}(u))({\check {R}}(uv)\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(v))=({\check {R}}(v)\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(uv))({\check {R}}(u)\otimes \mathbf {1} )} <p>In some cases, the determinant of R ( u ) {\displaystyle R(u)} can vanish at specific values of the spectral parameter u = u 0 {\displaystyle u=u_{0}} . Some R {\displaystyle R} matrices turn into a one dimensional projector at u = u 0 {\displaystyle u=u_{0}} . In this case a quantum determinant can be defined . </p> <h3>Example solutions of the parameter-dependent YBE</h3> <ul><li>A particularly simple class of parameter-dependent solutions can be obtained from solutions of the parameter-independent YBE satisfying R ˇ 2 = 1 {\displaystyle {\check {R}}^{2}=\mathbf {1} } , where the corresponding braid group representation is a permutation group representation. In this case, R ˇ ( u ) = 1 + u R ˇ {\displaystyle {\check {R}}(u)=\mathbf {1} +u{\check {R}}} (equivalently, R ( u ) = P + u P ∘ R ˇ {\displaystyle R(u)=P+uP\circ {\check {R}}} ) is a solution of the (additive) parameter-dependent YBE. In the case where R ˇ = P {\displaystyle {\check {R}}=P} and R ( u ) = P + u 1 {\displaystyle R(u)=P+u\mathbf {1} } , this gives the scattering matrix of the <a href="/facts/Heisenberg_model_(quantum)/8RPu8n5n">Heisenberg XXX spin chain</a>.</li></ul> <ul><li>The R {\displaystyle R} -matrices of the evaluation modules of the quantum group U q ( s l ^ ( 2 ) ) {\displaystyle U_{q}({\widehat {sl}}(2))} are given explicitly by the matrix</li></ul> R ˇ ( z ) = ( q z − q − 1 z − 1 q − q − 1 z − z − 1 z − z − 1 q − q − 1 q z − q − 1 z − 1 ) . {\displaystyle {\check {R}}(z)={\begin{pmatrix}qz-q^{-1}z^{-1}&&&\\&q-q^{-1}&z-z^{-1}&\\&z-z^{-1}&q-q^{-1}&\\&&&qz-q^{-1}z^{-1}\end{pmatrix}}.} <p>Then the parametrized Yang-Baxter equation (in braided form) with the multiplicative parameter is satisfied: </p> ( 1 ⊗ R ˇ ( z ) ) ( R ˇ ( z z ′ ) ⊗ 1 ) ( 1 ⊗ R ˇ ( z ′ ) ) = ( R ˇ ( z ′ ) ⊗ 1 ) ( 1 ⊗ R ˇ ( z z ′ ) ) ( R ˇ ( z ) ⊗ 1 ) {\displaystyle (\mathbf {1} \otimes {\check {R}}(z))({\check {R}}(zz')\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(z'))=({\check {R}}(z')\otimes \mathbf {1} )(\mathbf {1} \otimes {\check {R}}(zz'))({\check {R}}(z)\otimes \mathbf {1} )} <h3>Classification of solutions</h3> <p>There are broadly speaking three classes of solutions: rational, trigonometric and elliptic. These are related to <a href="/facts/Quantum_group/huNTuaXs">quantum groups</a> known as the <a href="/facts/Yangian/3EKUD0wQ">Yangian</a>, <a href="/facts/Affine_quantum_group/TBhseaW2">affine quantum groups</a> and <a href="/facts/Elliptic_algebra/I3kRcJbj">elliptic algebras</a> respectively. </p> <h2 id="set-theoretic-yangbaxter-equation">Set-theoretic Yang–Baxter equation</h2> <p>Set-theoretic solutions were studied by <a href="/facts/Vladimir_Drinfeld/xGhJ8NAv">Drinfeld</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> In this case, there is an R {\displaystyle R} -matrix invariant basis X {\displaystyle X} for the vector space V {\displaystyle V} in the sense that the R {\displaystyle R} -matrix maps the induced basis on V ⊗ V {\displaystyle V\otimes V} to itself. This then induces a map r : X × X → X × X {\displaystyle r:X\times X\rightarrow X\times X} given by restriction of the R {\displaystyle R} -matrix to the basis. The set-theoretic Yang–Baxter equation is then defined using the 'twisted' alternate form above, asserting ( i d × r ) ( r × i d ) ( i d × r ) = ( r × i d ) ( i d × r ) ( r × i d ) {\displaystyle (id\times r)(r\times id)(id\times r)=(r\times id)(id\times r)(r\times id)} as maps on X × X × X {\displaystyle X\times X\times X} . The equation can then be considered purely as an equation in the <a href="/facts/Category_of_sets/LF5ykYl0">category of sets</a>. </p> <h3>Examples</h3> <ul><li> R = i d {\displaystyle R=id} .</li> <li> R = τ {\displaystyle R=\tau } where τ ( u ⊗ v ) = v ⊗ u {\displaystyle \tau (u\otimes v)=v\otimes u} , the transposition map.</li> <li>If ( X , ◃ ) {\displaystyle (X,\triangleleft )} is a (right) shelf, then r ( x , y ) = ( y , x ◃ y ) {\displaystyle r(x,y)=(y,x\triangleleft y)} is a set-theoretic solution to the YBE.</li></ul> <h2 id="classical-yangbaxter-equation">Classical Yang–Baxter equation</h2> <p>Solutions to the classical YBE were studied and to some extent classified by <a href="/facts/Alexander_Belavin/0VKcvAa7">Belavin</a> and Drinfeld.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> Given a 'classical r {\displaystyle r} -matrix' r : V ⊗ V → V ⊗ V {\displaystyle r:V\otimes V\rightarrow V\otimes V} , which may also depend on a pair of arguments ( u , v ) {\displaystyle (u,v)} , the classical YBE is (suppressing parameters) [ r 12 , r 13 ] + [ r 12 , r 23 ] + [ r 13 , r 23 ] = 0. {\displaystyle [r_{12},r_{13}]+[r_{12},r_{23}]+[r_{13},r_{23}]=0.} This is quadratic in the r {\displaystyle r} -matrix, unlike the usual quantum YBE which is cubic in R {\displaystyle R} . </p><p>This equation emerges from so called quasi-classical solutions to the quantum YBE, in which the R {\displaystyle R} -matrix admits an asymptotic expansion in terms of an expansion parameter ℏ , {\displaystyle \hbar ,} R ℏ = I + ℏ r + O ( ℏ 2 ) . {\displaystyle R_{\hbar }=I+\hbar r+{\mathcal {O}}(\hbar ^{2}).} The classical YBE then comes from reading off the ℏ 2 {\displaystyle \hbar ^{2}} coefficient of the quantum YBE (and the equation trivially holds at orders ℏ 0 , ℏ {\displaystyle \hbar ^{0},\hbar } ). </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Lie_bialgebra/BVXwFTbW">Lie bialgebra</a></li> <li><a href="/facts/Yangian/3EKUD0wQ">Yangian</a></li> <li><a href="/facts/Reidemeister_move/0jQmU28C">Reidemeister move</a></li> <li><a href="/facts/Quasitriangular_Hopf_algebra/Y6acUkBu">Quasitriangular Hopf algebra</a></li> <li><a href="/facts/Yang%E2%80%93Baxter_operator/NXa06Q3S">Yang–Baxter operator</a></li></ul> <ul><li>H.-D. Doebner, J.-D. Hennig, eds, <i>Quantum groups, Proceedings of the 8th International Workshop on Mathematical Physics, Arnold Sommerfeld Institute, Clausthal, FRG, 1989</i>, Springer-Verlag Berlin, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-53503-9.</li> <li><a href="/facts/Vyjayanthi_Chari/49USnNSn">Vyjayanthi Chari</a> and Andrew Pressley, <i>A Guide to Quantum Groups</i>, (1994), Cambridge University Press, Cambridge <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-55884-0.</li> <li>Jacques H.H. Perk and Helen Au-Yang, "Yang–Baxter Equations", (2006), <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math-ph/0606053">math-ph/0606053</a>.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=Yang-Baxter_equation">"Yang-Baxter equation"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Jimbo, M. (1989). "Introduction to the Yang-Baxter Equation". International Journal of Modern Physics A. 4 (15). World Scientific: 3759–3777. Bibcode:1989IJMPA...4.3759J. doi:10.1142/S0217751X89001503. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>McGuire, J. B. (1964-05-01). "Study of Exactly Soluble One‐Dimensional N‐Body Problems". Journal of Mathematical Physics. 5 (5). The American Institute of Physics (AIP): 622–636. Bibcode:1964JMP.....5..622M. doi:10.1063/1.1704156. ISSN 0022-2488. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Yang, C. N. (1967-12-04). "Some Exact Results for the Many-Body Problem in one Dimension with Repulsive Delta-Function Interaction". Physical Review Letters. 19 (23). American Physical Society (APS): 1312–1315. Bibcode:1967PhRvL..19.1312Y. doi:10.1103/PhysRevLett.19.1312. ISSN 0031-9007. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Onsager, L. (1944-02-01). "Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition". Physical Review. 65 (3–4). 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ISSN 0003-4916. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Zamolodchikov, Alexander B. (1979). "Z4-symmetric factorized S-matrix in two space-time dimensions". Comm. Math. Phys. 69 (2). Elsevier: 165–178. Bibcode:1979CMaPh..69..165Z. doi:10.1007/BF01221446. ISSN 0003-4916. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Jucys, A. A. (1966). "On the Young operators of the symmetric group" (PDF). Lietuvos Fizikos Rinkinys. 6. Gos. 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