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41-489: Initially, they were considered a convenient tool to generate the solutions of the quantum Yang–Baxter equation . The center of the Yangian can be described by the quantum determinant . The Yangian is a degeneration of the quantum loop algebra (i.e. the quantum affine algebra at vanishing central charge). For any finite-dimensional semisimple Lie algebra a , Drinfeld defined an infinite-dimensional Hopf algebra Y (

82-671: A module 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

123-521: A ), called the Yangian of a . This Hopf algebra is a deformation of the universal enveloping algebra U ( a [ z ]) of the Lie algebra of polynomial loops of a given by explicit generators and relations. The relations can be encoded by identities involving a rational R -matrix . Replacing it with a trigonometric R -matrix, one arrives at affine quantum groups , defined in the same paper of Drinfeld. In

164-406: A matrix R {\displaystyle R} , acting on two out of three objects, satisfies where R ˇ {\displaystyle {\check {R}}} is R {\displaystyle R} followed by a swap of the two objects. In one-dimensional quantum systems, R {\displaystyle R} is the scattering matrix and if it satisfies

205-557: A multiplicative parameter, the Yang–Baxter equation is for all values of u {\displaystyle u} and v {\displaystyle v} . The braided forms read as: 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

246-413: A one dimensional projector at u = u 0 {\displaystyle u=u_{0}} . In this case a quantum determinant can be defined . Then the parametrized Yang-Baxter equation (in braided form) with the multiplicative parameter is satisfied: There are broadly speaking three classes of solutions: rational, trigonometric and elliptic. These are related to quantum groups known as

287-564: A scalar function, then R ′ {\displaystyle R'} also satisfies the Yang–Baxter equation. 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}

328-509: A symmetry group of one-dimensional exactly solvable models such as spin chains , Hubbard model and in models of one-dimensional relativistic quantum field theory . The most famous occurrence is in planar supersymmetric Yang–Mills theory in four dimensions, where Yangian structures appear on the level of symmetries of operators, and scattering amplitude as was discovered by Drummond, Henn and Plefka . Irreducible finite-dimensional representations of Yangians were parametrized by Drinfeld in

369-610: A way similar to the highest weight theory in the representation theory of semisimple Lie algebras. The role of the highest weight is played by a finite set of Drinfeld polynomials . Drinfeld also discovered a generalization of the classical Schur–Weyl duality between representations of general linear and symmetric groups that involves the Yangian of sl N and the degenerate affine Hecke algebra (graded Hecke algebra of type A, in George Lusztig 's terminology). Representations of Yangians have been extensively studied, but

410-456: Is a function of the scale-invariant ratio u / u ′ {\displaystyle u/u'} . 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

451-400: 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

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492-583: 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

533-459: 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 } ). Alexander Molev Alexander Ivanovich Molev ( Russian : Алекса́ндр Ива́нович Мо́лев ) FAA (born 1961) is a Russian - Australian mathematician. He completed his Ph.D. in 1986 under

574-419: The Yangian , affine quantum groups and elliptic algebras respectively. Set-theoretic solutions were studied by Drinfeld . 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

615-460: The category of sets . Solutions to the classical YBE were studied and to some extent classified by Belavin and Drinfeld. 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)} ,

656-586: The symmetric group in the work of A. A. Jucys in 1966. Let A {\displaystyle A} be a unital associative algebra . 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 tensor product A ⊗ A {\displaystyle A\otimes A} (here, u {\displaystyle u} and u ′ {\displaystyle u'} are

697-503: The Yang–Baxter equation are often constrained by requiring the R {\displaystyle R} matrix to be invariant under the action of a Lie group 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)} ,

738-466: The Yang–Baxter equation is for all values of u 1 {\displaystyle u_{1}} , u 2 {\displaystyle u_{2}} and u 3 {\displaystyle u_{3}} . 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

779-436: 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 R {\displaystyle R} corresponds to swapping two strands. Since one can swap three strands in two different ways, the Yang–Baxter equation enforces that both paths are the same. According to Jimbo , the Yang–Baxter equation (YBE) manifested itself in

820-1040: The alternate form is In the parameter-independent special case where R ˇ {\displaystyle {\check {R}}} does not depend on parameters, the equation reduces to and (if R {\displaystyle R} is invertible) a representation of the braid group , 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 braids , knots and links . Solutions to

861-427: The case of the general linear Lie algebra gl N , the Yangian admits a simpler description in terms of a single ternary (or RTT ) relation on the matrix generators due to Faddeev and coauthors. The Yangian Y( gl N ) is defined to be the algebra generated by elements t i j ( p ) {\displaystyle t_{ij}^{(p)}} with 1 ≤ i , j ≤ N and p ≥ 0, subject to

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902-418: The center of the Yangian. The twisted Yangian Y( gl 2N ), introduced by G. I. Olshansky, is the co-ideal generated by the coefficients of where σ is the involution of gl 2N given by G.I. Olshansky and I.Cherednik discovered that the Yangian of gl N is closely related with the branching properties of irreducible finite-dimensional representations of general linear algebras. In particular,

943-425: The classical Gelfand–Tsetlin construction of a basis in the space of such a representation has a natural interpretation in the language of Yangians, studied by M.Nazarov and V.Tarasov. Olshansky, Nazarov and Molev later discovered a generalization of this theory to other classical Lie algebras , based on the twisted Yangian. The Yangian appears as a symmetry group in different models in physics. Yangian appears as

984-400: 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

1025-421: 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

1066-511: The eight vertex model in 1972. Another line of development was the theory of factorized S-matrix in two dimensional quantum field theory. Zamolodchikov pointed out that the algebraic mechanics working here is the same as that in the Baxter's and others' works. The YBE has also manifested itself in a study of Young operators in the group algebra C [ S n ] {\displaystyle \mathbb {C} [S_{n}]} of

1107-721: 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} . 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

1148-403: 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

1189-642: 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 Let V {\displaystyle V} be

1230-765: 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} . The Yang–Baxter equation

1271-791: The parameters, which usually range over the real numbers ℝ in the case of an additive parameter, or over positive real numbers ℝ in the case of a multiplicative parameter). 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 homomorphisms ϕ i j : A ⊗ A → A ⊗ A ⊗ A {\displaystyle \phi _{ij}:A\otimes A\to A\otimes A\otimes A} determined by The general form of

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1312-514: 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: for all values of u {\displaystyle u} and v {\displaystyle v} . For

1353-478: The relations Defining t i j ( − 1 ) = δ i j {\displaystyle t_{ij}^{(-1)}=\delta _{ij}} , setting and introducing the R-matrix R ( z ) = I + z P on C ⊗ {\displaystyle \otimes } C , where P is the operator permuting the tensor factors, the above relations can be written more simply as

1394-641: The supervision of Alexandre Kirillov at Moscow State University . He was awarded the Australian Mathematical Society Medal in 2001 and became a Fellow of the Australian Academy of Science in 2019. Amongst other things, he has worked on Yangians and Lie algebras . He is currently a Professor in the School of Mathematics and Statistics , Faculty of Science , University of Sydney . This article about

1435-522: The tensor product A ⊗ A {\displaystyle A\otimes A} . The Yang–Baxter equation is 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)} . Often

1476-436: The ternary relation: The Yangian becomes a Hopf algebra with comultiplication Δ, counit ε and antipode s given by At special values of the spectral parameter ( z − w ) {\displaystyle (z-w)} , the R -matrix degenerates to a rank one projection. This can be used to define the quantum determinant of T ( z ) {\displaystyle T(z)} , which generates

1517-464: The theory is still under active development. Yang%E2%80%93Baxter equation 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 depends on the idea that in some scattering situations, particles may preserve their momentum while changing their quantum internal states. It states that

1558-466: The two-body problem, and determined it exactly. Here YBE arises as the consistency condition for the factorization. In statistical mechanics , 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 in 1944. The hunt for solvable lattice models has been actively pursued since then, culminating in Baxter's solution of

1599-448: The unital associative algebra is the algebra of endomorphisms of a vector space V {\displaystyle V} over a field k {\displaystyle k} , that is, A = End ( V ) {\displaystyle A={\text{End}}(V)} . With respect to a basis { e i } {\displaystyle \{e_{i}\}} of V {\displaystyle V} ,

1640-620: The usual quantum YBE which is cubic in R {\displaystyle R} . 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

1681-438: The works of J. B. McGuire in 1964 and C. N. Yang 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 Bethe's Ansatz techniques, they found that the scattering matrix factorized to that of

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