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Quantum groups and integrable lattice models UMN Math Physics Seminar October 14, 2013 Outline Lattice models: definitions and examples. Yang-Baxter equation and integrability. Quantum groups and their representations. Integrable lattice models from intertwining operators. UMN Math Physics Seminar Quantum groups and integrable lattice models Outline Lattice models: definitions and examples. Yang-Baxter equation and integrability. Quantum groups and their representations. Integrable lattice models from intertwining operators. UMN Math Physics Seminar Quantum groups and integrable lattice models Outline Lattice models: definitions and examples. Yang-Baxter equation and integrability. Quantum groups and their representations. Integrable lattice models from intertwining operators. UMN Math Physics Seminar Quantum groups and integrable lattice models Outline Lattice models: definitions and examples. Yang-Baxter equation and integrability. Quantum groups and their representations. Integrable lattice models from intertwining operators. UMN Math Physics Seminar Quantum groups and integrable lattice models Lattice models Consider a collection of “atoms” located at the vertices of a 2D-lattice ZM × ZN : Assumptions: each “atom” interacts only with its nearest neighbors; the energy of interaction depends only on the states of the bonds (the edges). bonds satisfy periodic (”toroidal“) boundary conditions. UMN Math Physics Seminar Quantum groups and integrable lattice models Lattice models Consider a collection of “atoms” located at the vertices of a 2D-lattice ZM × ZN : Assumptions: each “atom” interacts only with its nearest neighbors; the energy of interaction depends only on the states of the bonds (the edges). bonds satisfy periodic (”toroidal“) boundary conditions. UMN Math Physics Seminar Quantum groups and integrable lattice models Lattice models Consider a collection of “atoms” located at the vertices of a 2D-lattice ZM × ZN : Assumptions: each “atom” interacts only with its nearest neighbors; the energy of interaction depends only on the states of the bonds (the edges). bonds satisfy periodic (”toroidal“) boundary conditions. UMN Math Physics Seminar Quantum groups and integrable lattice models Lattice models Denote by E13,,24 the interaction energy of a single atom with the bonds in states 1 , . . . , 4 ∈ { 1, . . . , n }. | {z } possible states The state of the lattice is a map φ : bonds → {1, . . . , n} P 3 ,4 The energy of such a state is E(φ) := E1 ,2 atoms UMN Math Physics Seminar Quantum groups and integrable lattice models Lattice models Denote by E13,,24 the interaction energy of a single atom with the bonds in states 1 , . . . , 4 ∈ { 1, . . . , n }. | {z } possible states The state of the lattice is a map φ : bonds → {1, . . . , n} P 3 ,4 The energy of such a state is E(φ) := E1 ,2 atoms UMN Math Physics Seminar Quantum groups and integrable lattice models Partition function of a lattice model The partition function Z = ZM,N of such a system is Z= X exp(−βE(state)), where β = states 1 . kT We have exp(−βE(state)) = exp(−β X E13,,24 ) = atoms Y exp(−βE13,,24 ). atoms It what follows, it will be more convenient to work with the Boltzmann weights ,4 R13 , := exp(−βE13,,24 ) 2 rather than with energy terms. UMN Math Physics Seminar Quantum groups and integrable lattice models Partition function of a lattice model The partition function Z = ZM,N of such a system is Z= X exp(−βE(state)), where β = states 1 . kT We have exp(−βE(state)) = exp(−β X E13,,24 ) = atoms Y exp(−βE13,,24 ). atoms It what follows, it will be more convenient to work with the Boltzmann weights ,4 R13 , := exp(−βE13,,24 ) 2 rather than with energy terms. UMN Math Physics Seminar Quantum groups and integrable lattice models Partition function of a lattice model The partition function Z = ZM,N of such a system is Z= X exp(−βE(state)), where β = states 1 . kT We have exp(−βE(state)) = exp(−β X E13,,24 ) = atoms Y exp(−βE13,,24 ). atoms It what follows, it will be more convenient to work with the Boltzmann weights ,4 R13 , := exp(−βE13,,24 ) 2 rather than with energy terms. UMN Math Physics Seminar Quantum groups and integrable lattice models Partition function of a lattice model Standard properties of the partition function: P (the system is in a state with energy E) = 1 Z exp(−βE) If Q = Q(φ) is a function of states (an ”observable“), then its expectation value is hQi = 1 Z X Q(φ) exp(−βE(φ)). φ∈states Example 1 2 hEi = 1 Z P states ∂ E(φ) exp(−βE(φ)) = · · · = kT 2 ∂T ln Z The correlation function of edges j1 , . . . , jk is P hj1 . . . jk i = Z1 j1 · · · jk exp(−βE). states UMN Math Physics Seminar Quantum groups and integrable lattice models Partition function of a lattice model Standard properties of the partition function: P (the system is in a state with energy E) = 1 Z exp(−βE) If Q = Q(φ) is a function of states (an ”observable“), then its expectation value is hQi = 1 Z X Q(φ) exp(−βE(φ)). φ∈states Example 1 2 hEi = 1 Z P states ∂ E(φ) exp(−βE(φ)) = · · · = kT 2 ∂T ln Z The correlation function of edges j1 , . . . , jk is P hj1 . . . jk i = Z1 j1 · · · jk exp(−βE). states UMN Math Physics Seminar Quantum groups and integrable lattice models Partition function of a lattice model Standard properties of the partition function: P (the system is in a state with energy E) = 1 Z exp(−βE) If Q = Q(φ) is a function of states (an ”observable“), then its expectation value is hQi = 1 Z X Q(φ) exp(−βE(φ)). φ∈states Example 1 2 hEi = 1 Z P states ∂ E(φ) exp(−βE(φ)) = · · · = kT 2 ∂T ln Z The correlation function of edges j1 , . . . , jk is P hj1 . . . jk i = Z1 j1 · · · jk exp(−βE). states UMN Math Physics Seminar Quantum groups and integrable lattice models Partition function of a lattice model Standard properties of the partition function: P (the system is in a state with energy E) = 1 Z exp(−βE) If Q = Q(φ) is a function of states (an ”observable“), then its expectation value is hQi = 1 Z X Q(φ) exp(−βE(φ)). φ∈states Example 1 2 hEi = 1 Z P states ∂ E(φ) exp(−βE(φ)) = · · · = kT 2 ∂T ln Z The correlation function of edges j1 , . . . , jk is P hj1 . . . jk i = Z1 j1 · · · jk exp(−βE). states UMN Math Physics Seminar Quantum groups and integrable lattice models Partition function of a lattice model Standard properties of the partition function: P (the system is in a state with energy E) = 1 Z exp(−βE) If Q = Q(φ) is a function of states (an ”observable“), then its expectation value is hQi = 1 Z X Q(φ) exp(−βE(φ)). φ∈states Example 1 2 hEi = 1 Z P states ∂ E(φ) exp(−βE(φ)) = · · · = kT 2 ∂T ln Z The correlation function of edges j1 , . . . , jk is P hj1 . . . jk i = Z1 j1 · · · jk exp(−βE). states UMN Math Physics Seminar Quantum groups and integrable lattice models Examples of lattice models 1) Eight-vertex model. Characteristics: two bond states (we denote them by + and −). only 8 out of 16 possible vertex configurations are allowed: Here, a, b, c, d are Boltzmann weights of the corresponding configurations. UMN Math Physics Seminar Quantum groups and integrable lattice models Examples of lattice models 2) Six-vertex model (two-dimensional ice). Characteristics: two bond states (we denote them by + and −). only 6 out of 16 possible vertex configurations are allowed: UMN Math Physics Seminar Quantum groups and integrable lattice models Lattice models To solve a model (a lattice model, in our context) means to find an explicit formula for Z = ZM,N , its thermodynamical limit lim ZM,N or thermodynamical limit per site M,N →∞ 1 lim (ZM,N ) M N M,N →∞ Example Eight- and six-vertex models are solvable (R.Baxter, 1971). UMN Math Physics Seminar Quantum groups and integrable lattice models Transfer matrix Z= X exp(−βE(state)) = states X Y ,4 R13 , 2 states atoms The contribution of a single column of the lattice to the partition function is X 0 ...0 ν1 0N ν 0 ν 0 T11...NN = Rν12 11 Rν23 22 . . . RνN N ν1 ,...,νN ∈{+,−} UMN Math Physics Seminar Quantum groups and integrable lattice models Transfer matrix Z= X exp(−βE(state)) = states X Y ,4 R13 , 2 states atoms The contribution of a single column of the lattice to the partition function is X 0 ...0 ν1 0N ν 0 ν 0 T11...NN = Rν12 11 Rν23 22 . . . RνN N ν1 ,...,νN ∈{+,−} UMN Math Physics Seminar Quantum groups and integrable lattice models Transfer matrix 0 ...0 We let V = Chv+ , v− i and regard T11...NN ’s as coefficients of a linear operator (”the transfer matrix “) T : V ⊗N → V ⊗N X v1 ⊗ · · · ⊗ vN 7→ 0 ...0 T11...NN v01 ⊗ · · · ⊗ v0N 0i ∈{+,−} UMN Math Physics Seminar Quantum groups and integrable lattice models Transfer matrix Observation The coefficients of T ◦ T capture the contributions from two consecutive columns; the coefficients of T ◦ T ◦ T do this for three columns and so on. Due to the periodic boundary conditions, we have the following Proposition ZM,N = tr(T M ) UMN Math Physics Seminar Quantum groups and integrable lattice models Transfer matrix Observation The coefficients of T ◦ T capture the contributions from two consecutive columns; the coefficients of T ◦ T ◦ T do this for three columns and so on. Due to the periodic boundary conditions, we have the following Proposition ZM,N = tr(T M ) UMN Math Physics Seminar Quantum groups and integrable lattice models Transfer matrix Let λ1 ≥ λ2 ≥ . . . be the eigenvalues of T ZM,N = tr(T M ) = λM 1 1+ λ2 λ1 ! M + ... . So Z ∼ λM 1 for M >> 0. Solving the six-vertex model Solving the eigenvalue problem for T The eigenvalue problem will simplify once we find a (large) family of operators commuting with T . Goal Construct a family of operators commuting with T . UMN Math Physics Seminar Quantum groups and integrable lattice models Transfer matrix Let λ1 ≥ λ2 ≥ . . . be the eigenvalues of T ZM,N = tr(T M ) = λM 1 1+ λ2 λ1 ! M + ... . So Z ∼ λM 1 for M >> 0. Solving the six-vertex model Solving the eigenvalue problem for T The eigenvalue problem will simplify once we find a (large) family of operators commuting with T . Goal Construct a family of operators commuting with T . UMN Math Physics Seminar Quantum groups and integrable lattice models Transfer matrix Let λ1 ≥ λ2 ≥ . . . be the eigenvalues of T ZM,N = tr(T M ) = λM 1 1+ λ2 λ1 ! M + ... . So Z ∼ λM 1 for M >> 0. Solving the six-vertex model Solving the eigenvalue problem for T The eigenvalue problem will simplify once we find a (large) family of operators commuting with T . Goal Construct a family of operators commuting with T . UMN Math Physics Seminar Quantum groups and integrable lattice models Transfer matrix Let λ1 ≥ λ2 ≥ . . . be the eigenvalues of T ZM,N = tr(T M ) = λM 1 1+ λ2 λ1 ! M + ... . So Z ∼ λM 1 for M >> 0. Solving the six-vertex model Solving the eigenvalue problem for T The eigenvalue problem will simplify once we find a (large) family of operators commuting with T . Goal Construct a family of operators commuting with T . UMN Math Physics Seminar Quantum groups and integrable lattice models Transfer matrix Let λ1 ≥ λ2 ≥ . . . be the eigenvalues of T ZM,N = tr(T M ) = λM 1 1+ λ2 λ1 ! M + ... . So Z ∼ λM 1 for M >> 0. Solving the six-vertex model Solving the eigenvalue problem for T The eigenvalue problem will simplify once we find a (large) family of operators commuting with T . Goal Construct a family of operators commuting with T . UMN Math Physics Seminar Quantum groups and integrable lattice models Transfer matrix Remark The six-vertex model is equivalent (in a rather precise sense - relating the transfer matrix to the Hamiltonian) to the Heisenberg XXZ-chain model. This is an example of the quantum/statistical correspondence. (d + 1) − dimensional d − dimensional classical statistical ! quantum model model Under this correspondence, the transfer matrix T is analogous to the infinitesimal time evolution operator e−H∆τ . UMN Math Physics Seminar Quantum groups and integrable lattice models Transfer matrix Remark The six-vertex model is equivalent (in a rather precise sense - relating the transfer matrix to the Hamiltonian) to the Heisenberg XXZ-chain model. This is an example of the quantum/statistical correspondence. (d + 1) − dimensional d − dimensional classical statistical ! quantum model model Under this correspondence, the transfer matrix T is analogous to the infinitesimal time evolution operator e−H∆τ . UMN Math Physics Seminar Quantum groups and integrable lattice models Transfer matrix Remark The six-vertex model is equivalent (in a rather precise sense - relating the transfer matrix to the Hamiltonian) to the Heisenberg XXZ-chain model. This is an example of the quantum/statistical correspondence. (d + 1) − dimensional d − dimensional classical statistical ! quantum model model Under this correspondence, the transfer matrix T is analogous to the infinitesimal time evolution operator e−H∆τ . UMN Math Physics Seminar Quantum groups and integrable lattice models Transfer matrix Remark The six-vertex model is equivalent (in a rather precise sense - relating the transfer matrix to the Hamiltonian) to the Heisenberg XXZ-chain model. This is an example of the quantum/statistical correspondence. (d + 1) − dimensional d − dimensional classical statistical ! quantum model model Under this correspondence, the transfer matrix T is analogous to the infinitesimal time evolution operator e−H∆τ . UMN Math Physics Seminar Quantum groups and integrable lattice models R-matrix Working in the same vector space V = Chv+ , v− i, we define R:V ⊗V →V ⊗V X v1 ⊗ v2 7→ R01 02 v01 ⊗ v02 01 ,02 ∈{+,−} 1 2 This operator (”R-matrix “) captures contributions of a single vertex to the partition function. Consider an (N + 1)-fold tensor product V0 ⊗ V1 ⊗ · · · ⊗ VN (Vi = V ) and let Rij be the operator acting on the Vi ⊗ Vj component of this product as R and as identity on any other Vl . UMN Math Physics Seminar Quantum groups and integrable lattice models R-matrix Working in the same vector space V = Chv+ , v− i, we define R:V ⊗V →V ⊗V X v1 ⊗ v2 7→ R01 02 v01 ⊗ v02 01 ,02 ∈{+,−} 1 2 Consider an (N + 1)-fold tensor product V0 ⊗ V1 ⊗ · · · ⊗ VN (Vi = V ) and let Rij be the operator acting on the Vi ⊗ Vj component of this product as R and as identity on any other Vl . UMN Math Physics Seminar Quantum groups and integrable lattice models R-matrix Working in the same vector space V = Chv+ , v− i, we define R:V ⊗V →V ⊗V X v1 ⊗ v2 7→ R01 02 v01 ⊗ v02 01 ,02 ∈{+,−} 1 2 Consider an (N + 1)-fold tensor product V0 ⊗ V1 ⊗ · · · ⊗ VN (Vi = V ) and let Rij be the operator acting on the Vi ⊗ Vj component of this product as R and as identity on any other Vl . R0N . . . R02 R01 : V0 ⊗ V1 ⊗ · · · ⊗ VN → V0 ⊗ V1 ⊗ · · · ⊗ VN UMN Math Physics Seminar Quantum groups and integrable lattice models R-matrix Working in the same vector space V = Chv+ , v− i, we define R:V ⊗V →V ⊗V X v1 ⊗ v2 7→ R01 02 v01 ⊗ v02 01 ,02 ∈{+,−} 1 2 Consider an (N + 1)-fold tensor product V0 ⊗ V1 ⊗ · · · ⊗ VN (Vi = V ) and let Rij be the operator acting on the Vi ⊗ Vj component of this product as R and as identity on any other Vl . R0N . . . R02 R01 : V0 ⊗ (V1 ⊗ · · · ⊗ VN ) → V0 ⊗ (V1 ⊗ · · · ⊗ VN ) UMN Math Physics Seminar Quantum groups and integrable lattice models R-matrix Working in the same vector space V = Chv+ , v− i, we define R:V ⊗V →V ⊗V X v1 ⊗ v2 7→ R01 02 v01 ⊗ v02 01 ,02 ∈{+,−} 1 2 Consider an (N + 1)-fold tensor product V0 ⊗ V1 ⊗ · · · ⊗ VN (Vi = V ) and let Rij be the operator acting on the Vi ⊗ Vj component of this product as R and as identity on any other Vl . R0N . . . R02 R01 : V0 ⊗ (V1 ⊗ · · · ⊗ VN ) → V0 ⊗ (V1 ⊗ · · · ⊗ VN ) A B L= C D | {z } 00 monodromy matrix“ UMN Math Physics Seminar Quantum groups and integrable lattice models R-matrix Working in the same vector space V = Chv+ , v− i, we define R:V ⊗V →V ⊗V X v1 ⊗ v2 7→ R01 02 v01 ⊗ v02 01 ,02 ∈{+,−} 1 2 Consider an (N + 1)-fold tensor product V0 ⊗ V1 ⊗ · · · ⊗ VN (Vi = V ) and let Rij be the operator acting on the Vi ⊗ Vj component of this product as R and as identity on any other Vl . R0N . . . R02 R01 : V0 ⊗ (V1 ⊗ · · · ⊗ VN ) → V0 ⊗ (V1 ⊗ · · · ⊗ VN ) A B L= C D | {z } 00 monodromy matrix“ trV0 (R0N . . . R02 R01 ) = A + D UMN Math Physics Seminar Quantum groups and integrable lattice models R-matrix Working in the same vector space V = Chv+ , v− i, we define R:V ⊗V →V ⊗V X v1 ⊗ v2 7→ R01 02 v01 ⊗ v02 01 ,02 ∈{+,−} 1 2 Consider an (N + 1)-fold tensor product V0 ⊗ V1 ⊗ · · · ⊗ VN (Vi = V ) and let Rij be the operator acting on the Vi ⊗ Vj component of this product as R and as identity on any other Vl . R0N . . . R02 R01 : V0 ⊗ (V1 ⊗ · · · ⊗ VN ) → V0 ⊗ (V1 ⊗ · · · ⊗ VN ) Proposition T = trV0 (R0N . . . R02 R01 ) UMN Math Physics Seminar Quantum groups and integrable lattice models R-matrix Example For the six-vertex model, the R-matrix (in the appropriate basis of V ⊗ V ) is d 0 0 0 0 b c 0 (1) R= 0 c b 0 0 0 0 d Theorem Let R and R0 be matrices of the form (1). Define 0 . . . R0 ). T = tr(R0N . . . R01 ) and T 0 = tr(R0N 01 If there is a matrix R00 of the form (1) such that 00 0 0 00 R12 R13 R23 = R23 R13 R12 (on V ⊗ V ⊗ V ) (2) then [T, T 0 ] = 0. UMN Physics Quantum groups and integrable lattice models Equation (2) is Math known asSeminar the quantum Yang-Baxter equation. R-matrix Example For the six-vertex model, the R-matrix (in the appropriate basis of V ⊗ V ) is d 0 0 0 0 b c 0 (1) R= 0 c b 0 0 0 0 d Theorem Let R and R0 be matrices of the form (1). Define 0 . . . R0 ). T = tr(R0N . . . R01 ) and T 0 = tr(R0N 01 If there is a matrix R00 of the form (1) such that 00 0 0 00 R12 R13 R23 = R23 R13 R12 (on V ⊗ V ⊗ V ) (2) then [T, T 0 ] = 0. UMN Physics Quantum groups and integrable lattice models Equation (2) is Math known asSeminar the quantum Yang-Baxter equation. R-matrices and commuting operators Theorem Let R and R0 be matrices of the form (1). Define 0 . . . R0 ). T = trV0 (R0N . . . R01 ) and T 0 = trV00 (R0N 01 00 If there is a matrix R of the form (1) such that 00 0 0 00 R12 R13 R23 = R23 R13 R12 (on V ⊗ V ⊗ V ) (2) then [T, T 0 ] = 0. Plan of the proof. Repeatedly using QYBE, show that 00−1 00 0 R12 L LR12 = LL0 , where L = R0N . . . R01 , L0 = R00 0 N . . . R00 0 1 are the monodromy operators acting on V0 ⊗ V00 ⊗ V1 ⊗ . . . VN . Take the trace of the above identity over V0 ⊗ V00 and use the previous proposition. UMN Math Physics Seminar Quantum groups and integrable lattice models R-matrices and commuting operators Theorem Let R and R0 be matrices of the form (1). Define 0 . . . R0 ). T = trV0 (R0N . . . R01 ) and T 0 = trV00 (R0N 01 00 If there is a matrix R of the form (1) such that 00 0 0 00 R12 R13 R23 = R23 R13 R12 (on V ⊗ V ⊗ V ) (2) then [T, T 0 ] = 0. Plan of the proof. Repeatedly using QYBE, show that 00−1 00 0 R12 L LR12 = LL0 , where L = R0N . . . R01 , L0 = R00 0 N . . . R00 0 1 are the monodromy operators acting on V0 ⊗ V00 ⊗ V1 ⊗ . . . VN . Take the trace of the above identity over V0 ⊗ V00 and use the previous proposition. UMN Math Physics Seminar Quantum groups and integrable lattice models R-matrices and commuting operators Theorem Let R and R0 be matrices of the form (1). Define 0 . . . R0 ). T = trV0 (R0N . . . R01 ) and T 0 = trV00 (R0N 01 00 If there is a matrix R of the form (1) such that 00 0 0 00 R12 R13 R23 = R23 R13 R12 (on V ⊗ V ⊗ V ) (2) then [T, T 0 ] = 0. Plan of the proof. Repeatedly using QYBE, show that 00−1 00 0 R12 L LR12 = LL0 , where L = R0N . . . R01 , L0 = R00 0 N . . . R00 0 1 are the monodromy operators acting on V0 ⊗ V00 ⊗ V1 ⊗ . . . VN . Take the trace of the above identity over V0 ⊗ V00 and use the previous proposition. UMN Math Physics Seminar Quantum groups and integrable lattice models Integrability Definition A hamiltonian dynamical system is said to be completely integrable if it has the maximal possible number of conserved quantities in involution (by Liouville, it’s 21 dim(phase space)). By analogy, we would call a lattice model integrable if it admits a “large” family of operators commuting with each other and with T . Due to the previous theorem, that can be formalized as follows: Definition A lattice model is integrable if there is a family of R-matrices depending on parameters λ, µ, ν such that for any µ, ν, there is a λ such that R12 (λ)R13 (µ)R23 (ν) = R23 (ν)R13 (µ)R12 (λ) UMN Math Physics Seminar Quantum groups and integrable lattice models Integrability Definition A hamiltonian dynamical system is said to be completely integrable if it has the maximal possible number of conserved quantities in involution (by Liouville, it’s 21 dim(phase space)). By analogy, we would call a lattice model integrable if it admits a “large” family of operators commuting with each other and with T . Due to the previous theorem, that can be formalized as follows: Definition A lattice model is integrable if there is a family of R-matrices depending on parameters λ, µ, ν such that for any µ, ν, there is a λ such that R12 (λ)R13 (µ)R23 (ν) = R23 (ν)R13 (µ)R12 (λ) UMN Math Physics Seminar Quantum groups and integrable lattice models Integrability Definition A hamiltonian dynamical system is said to be completely integrable if it has the maximal possible number of conserved quantities in involution (by Liouville, it’s 21 dim(phase space)). By analogy, we would call a lattice model integrable if it admits a “large” family of operators commuting with each other and with T . Due to the previous theorem, that can be formalized as follows: Definition A lattice model is integrable if there is a family of R-matrices depending on parameters λ, µ, ν such that for any µ, ν, there is a λ such that R12 (λ)R13 (µ)R23 (ν) = R23 (ν)R13 (µ)R12 (λ) UMN Math Physics Seminar Quantum groups and integrable lattice models Quantum integrability In general, a QYBE is a system of 64 non-linear algebraic equations with 16 variables. In case of the six-vertex model, it boils down to three equations that can be solved explicitly. A family of solutions is given by ρsh(η + u) 0 0 0 0 ρsh(u) ρsh(η) 0 R= 0 ρsh(η) ρsh(u) 0 0 0 0 ρsh(η + u) Question Is there a systematic way of constructing R-matrices (=integrable lattice models) for the cases other than six- or eight-vertex models? Answer(V.Drinfeld, M.Jimbo and others) Yes. UMN Math Physics Seminar Quantum groups and integrable lattice models Quantum integrability In general, a QYBE is a system of 64 non-linear algebraic equations with 16 variables. In case of the six-vertex model, it boils down to three equations that can be solved explicitly. A family of solutions is given by ρsh(η + u) 0 0 0 0 ρsh(u) ρsh(η) 0 R= 0 ρsh(η) ρsh(u) 0 0 0 0 ρsh(η + u) Question Is there a systematic way of constructing R-matrices (=integrable lattice models) for the cases other than six- or eight-vertex models? Answer(V.Drinfeld, M.Jimbo and others) Yes. UMN Math Physics Seminar Quantum groups and integrable lattice models Quantum integrability In general, a QYBE is a system of 64 non-linear algebraic equations with 16 variables. In case of the six-vertex model, it boils down to three equations that can be solved explicitly. A family of solutions is given by ρsh(η + u) 0 0 0 0 ρsh(u) ρsh(η) 0 R= 0 ρsh(η) ρsh(u) 0 0 0 0 ρsh(η + u) Similarly, there is a family of solutions of the QYBE for the eight-vertex model given in terms of elliptic functions. Question Is there a systematic way of constructing R-matrices (=integrable lattice models) for the cases other than six- or eight-vertex models? UMN Math Physics Seminar and Quantum groups and integrable lattice models Answer(V.Drinfeld, M.Jimbo others) Yes. Quantum integrability In general, a QYBE is a system of 64 non-linear algebraic equations with 16 variables. In case of the six-vertex model, it boils down to three equations that can be solved explicitly. A family of solutions is given by ρsh(η + u) 0 0 0 0 ρsh(u) ρsh(η) 0 R= 0 ρsh(η) ρsh(u) 0 0 0 0 ρsh(η + u) Question Is there a systematic way of constructing R-matrices (=integrable lattice models) for the cases other than six- or eight-vertex models? Answer(V.Drinfeld, M.Jimbo and others) Yes. UMN Math Physics Seminar Quantum groups and integrable lattice models Quantum integrability In general, a QYBE is a system of 64 non-linear algebraic equations with 16 variables. In case of the six-vertex model, it boils down to three equations that can be solved explicitly. A family of solutions is given by ρsh(η + u) 0 0 0 0 ρsh(u) ρsh(η) 0 R= 0 ρsh(η) ρsh(u) 0 0 0 0 ρsh(η + u) Question Is there a systematic way of constructing R-matrices (=integrable lattice models) for the cases other than six- or eight-vertex models? Answer(V.Drinfeld, M.Jimbo and others) Yes. UMN Math Physics Seminar Quantum groups and integrable lattice models Hopf algebras and quantum groups Definition A coalgebra over a commutative ring k is a k-module A equipped with a comultiplication map ∆ : A → A ⊗ A and a counit A → k subject to coassociativity and counitality conditions (“diagrammatical” duals of the usual associativity and unit conditions). Definition A Hopf algebra over k is a k-module A such that A is both unital algebra and coalgebra; (co)multiplication and (co)unit are homomorphism of coalgebras (algebras); there is a bijective k-linear map S, called the antipode, such that µ ⊗ (S ⊗ id) ⊗ ∆ = i ⊗ and µ ⊗ (id ⊗ S) ⊗ ∆ = i ⊗ . UMN Math Physics Seminar Quantum groups and integrable lattice models Hopf algebras and quantum groups Definition A coalgebra over a commutative ring k is a k-module A equipped with a comultiplication map ∆ : A → A ⊗ A and a counit A → k subject to coassociativity and counitality conditions (“diagrammatical” duals of the usual associativity and unit conditions). Definition A Hopf algebra over k is a k-module A such that A is both unital algebra and coalgebra; (co)multiplication and (co)unit are homomorphism of coalgebras (algebras); there is a bijective k-linear map S, called the antipode, such that µ ⊗ (S ⊗ id) ⊗ ∆ = i ⊗ and µ ⊗ (id ⊗ S) ⊗ ∆ = i ⊗ . UMN Math Physics Seminar Quantum groups and integrable lattice models Hopf algebras and quantum groups Definition A coalgebra over a commutative ring k is a k-module A equipped with a comultiplication map ∆ : A → A ⊗ A and a counit A → k subject to coassociativity and counitality conditions (“diagrammatical” duals of the usual associativity and unit conditions). Definition A Hopf algebra over k is a k-module A such that A is both unital algebra and coalgebra; (co)multiplication and (co)unit are homomorphism of coalgebras (algebras); there is a bijective k-linear map S, called the antipode, such that µ ⊗ (S ⊗ id) ⊗ ∆ = i ⊗ and µ ⊗ (id ⊗ S) ⊗ ∆ = i ⊗ . UMN Math Physics Seminar Quantum groups and integrable lattice models Hopf algebras and quantum groups Definition A coalgebra over a commutative ring k is a k-module A equipped with a comultiplication map ∆ : A → A ⊗ A and a counit A → k subject to coassociativity and counitality conditions (“diagrammatical” duals of the usual associativity and unit conditions). Definition A Hopf algebra over k is a k-module A such that A is both unital algebra and coalgebra; (co)multiplication and (co)unit are homomorphism of coalgebras (algebras); there is a bijective k-linear map S, called the antipode, such that µ ⊗ (S ⊗ id) ⊗ ∆ = i ⊗ and µ ⊗ (id ⊗ S) ⊗ ∆ = i ⊗ . UMN Math Physics Seminar Quantum groups and integrable lattice models Hopf algebras and quantum groups Definition A coalgebra over a commutative ring k is a k-module A equipped with a comultiplication map ∆ : A → A ⊗ A and a counit A → k subject to coassociativity and counitality conditions (“diagrammatical” duals of the usual associativity and unit conditions). Definition A Hopf algebra over k is a k-module A such that A is both unital algebra and coalgebra; (co)multiplication and (co)unit are homomorphism of coalgebras (algebras); there is a bijective k-linear map S, called the antipode, such that µ ⊗ (S ⊗ id) ⊗ ∆ = i ⊗ and µ ⊗ (id ⊗ S) ⊗ ∆ = i ⊗ . UMN Math Physics Seminar Quantum groups and integrable lattice models Hopf algebras and quantum groups Example 1 For a group G, a group algebra k[G] equipped with a coproduct ∆(g) = g ⊗ g and an antipode S(g) = g −1 is a Hopf algebra. 2 Let G be a finite group. The algebra of k-functions F(G) on G is a commutative Hopf algebra with a comultiplication is ∆(f )(g1 , g2 ) = f (g1 g2 ) and an antipode S(f )(g) = f (g −1 ). A variation of this construction exists for compact topological groups. 3 Let g be a Lie algebra over a field k. The universal enveloping algebra U (g) acquires a structure of a Hopf algebra via ∆(x) = x ⊗ 1 + 1 ⊗ x, S(x) = −x, (x) = 0 for x ∈ g. UMN Math Physics Seminar Quantum groups and integrable lattice models Hopf algebras and quantum groups Example 1 For a group G, a group algebra k[G] equipped with a coproduct ∆(g) = g ⊗ g and an antipode S(g) = g −1 is a Hopf algebra. 2 Let G be a finite group. The algebra of k-functions F(G) on G is a commutative Hopf algebra with a comultiplication is ∆(f )(g1 , g2 ) = f (g1 g2 ) and an antipode S(f )(g) = f (g −1 ). A variation of this construction exists for compact topological groups. 3 Let g be a Lie algebra over a field k. The universal enveloping algebra U (g) acquires a structure of a Hopf algebra via ∆(x) = x ⊗ 1 + 1 ⊗ x, S(x) = −x, (x) = 0 for x ∈ g. UMN Math Physics Seminar Quantum groups and integrable lattice models Hopf algebras and quantum groups Example 1 For a group G, a group algebra k[G] equipped with a coproduct ∆(g) = g ⊗ g and an antipode S(g) = g −1 is a Hopf algebra. 2 Let G be a finite group. The algebra of k-functions F(G) on G is a commutative Hopf algebra with a comultiplication is ∆(f )(g1 , g2 ) = f (g1 g2 ) and an antipode S(f )(g) = f (g −1 ). A variation of this construction exists for compact topological groups. 3 Let g be a Lie algebra over a field k. The universal enveloping algebra U (g) acquires a structure of a Hopf algebra via ∆(x) = x ⊗ 1 + 1 ⊗ x, S(x) = −x, (x) = 0 for x ∈ g. UMN Math Physics Seminar Quantum groups and integrable lattice models Hopf algebras and quantum groups Example 1 For a group G, a group algebra k[G] equipped with a coproduct ∆(g) = g ⊗ g and an antipode S(g) = g −1 is a Hopf algebra. 2 Let G be a finite group. The algebra of k-functions F(G) on G is a commutative Hopf algebra with a comultiplication is ∆(f )(g1 , g2 ) = f (g1 g2 ) and an antipode S(f )(g) = f (g −1 ). A variation of this construction exists for compact topological groups. 3 Let g be a Lie algebra over a field k. The universal enveloping algebra U (g) acquires a structure of a Hopf algebra via ∆(x) = x ⊗ 1 + 1 ⊗ x, S(x) = −x, (x) = 0 for x ∈ g. UMN Math Physics Seminar Quantum groups and integrable lattice models Hopf algebras A representation of a Hopf algebra A is a module V over the algebra A. A nice property: two representations V, W of a Hopf algebra A can be tensor-multiplied: a · (v ⊗ w) = ∆(a).(v ⊗ w) UMN Math Physics Seminar Quantum groups and integrable lattice models Hopf algebras A representation of a Hopf algebra A is a module V over the algebra A. A nice property: two representations V, W of a Hopf algebra A can be tensor-multiplied: a · (v ⊗ w) = ∆(a).(v ⊗ w) UMN Math Physics Seminar Quantum groups and integrable lattice models Hopf algebras and quantum groups Definition A Hopf algebra A is called cocommutative if ∆op := σ ⊗ ∆ = ∆, where σ is the transposition v ⊗ u 7→ u ⊗ v. A Hopf algebra A is called almost cocommutative if there exists an invertible element R ∈ A ⊗ A such that ∆op = R∆R−1 . A quasitriangular Hopf algebra (=a quantum group) is an almost cocommutative Hopf algebra (A, R) such that (∆ ⊗ id)(R) = R13 R23 , (id ⊗ ∆)(R) = R13 R12 where R13 := (σ ⊗ id)(R), R12 := R ⊗ 1, R23 = 1 ⊗ R. UMN Math Physics Seminar Quantum groups and integrable lattice models Hopf algebras and quantum groups Definition A Hopf algebra A is called cocommutative if ∆op := σ ⊗ ∆ = ∆, where σ is the transposition v ⊗ u 7→ u ⊗ v. A Hopf algebra A is called almost cocommutative if there exists an invertible element R ∈ A ⊗ A such that ∆op = R∆R−1 . A quasitriangular Hopf algebra (=a quantum group) is an almost cocommutative Hopf algebra (A, R) such that (∆ ⊗ id)(R) = R13 R23 , (id ⊗ ∆)(R) = R13 R12 where R13 := (σ ⊗ id)(R), R12 := R ⊗ 1, R23 = 1 ⊗ R. UMN Math Physics Seminar Quantum groups and integrable lattice models Hopf algebras and quantum groups Definition A Hopf algebra A is called cocommutative if ∆op := σ ⊗ ∆ = ∆, where σ is the transposition v ⊗ u 7→ u ⊗ v. A Hopf algebra A is called almost cocommutative if there exists an invertible element R ∈ A ⊗ A such that ∆op = R∆R−1 . A quasitriangular Hopf algebra (=a quantum group) is an almost cocommutative Hopf algebra (A, R) such that (∆ ⊗ id)(R) = R13 R23 , (id ⊗ ∆)(R) = R13 R12 where R13 := (σ ⊗ id)(R), R12 := R ⊗ 1, R23 = 1 ⊗ R. UMN Math Physics Seminar Quantum groups and integrable lattice models Hopf algebras and quantum groups Definition A Hopf algebra A is called cocommutative if ∆op := σ ⊗ ∆ = ∆, where σ is the transposition v ⊗ u 7→ u ⊗ v. A Hopf algebra A is called almost cocommutative if there exists an invertible element R ∈ A ⊗ A such that ∆op = R∆R−1 . A quasitriangular Hopf algebra (=a quantum group) is an almost cocommutative Hopf algebra (A, R) such that (∆ ⊗ id)(R) = R13 R23 , (id ⊗ ∆)(R) = R13 R12 where R13 := (σ ⊗ id)(R), R12 := R ⊗ 1, R23 = 1 ⊗ R. UMN Math Physics Seminar Quantum groups and integrable lattice models Hopf algebras and quantum groups What is the meaning of the quasitriangular condition? It’s mainly due to the following Theorem Let A be a Hopf algebra. Then the category of A-modules (=representations of A) is braided(=has intertwiners) if and only if A is quasitriangular. What quantum groups have to do with QYBE? Proposition Let (A, R) be a quasitriangular Hopf algebra. Then the following form of QYBE holds in A⊗3 : R12 R13 R23 = R23 R13 R12 An upshot: finite-dimensional representations of a quantum group give rise to an R-matrices. For this reason, R is sometimes called a universal R-matrix. UMN Math Physics Seminar Quantum groups and integrable lattice models Hopf algebras and quantum groups What is the meaning of the quasitriangular condition? It’s mainly due to the following Theorem Let A be a Hopf algebra. Then the category of A-modules (=representations of A) is braided(=has intertwiners) if and only if A is quasitriangular. What quantum groups have to do with QYBE? Proposition Let (A, R) be a quasitriangular Hopf algebra. Then the following form of QYBE holds in A⊗3 : R12 R13 R23 = R23 R13 R12 An upshot: finite-dimensional representations of a quantum group give rise to an R-matrices. For this reason, R is sometimes called a universal R-matrix. UMN Math Physics Seminar Quantum groups and integrable lattice models Hopf algebras and quantum groups What is the meaning of the quasitriangular condition? It’s mainly due to the following Theorem Let A be a Hopf algebra. Then the category of A-modules (=representations of A) is braided(=has intertwiners) if and only if A is quasitriangular. What quantum groups have to do with QYBE? Proposition Let (A, R) be a quasitriangular Hopf algebra. Then the following form of QYBE holds in A⊗3 : R12 R13 R23 = R23 R13 R12 An upshot: finite-dimensional representations of a quantum group give rise to an R-matrices. For this reason, R is sometimes called a universal R-matrix. UMN Math Physics Seminar Quantum groups and integrable lattice models Hopf algebras and quantum groups What is the meaning of the quasitriangular condition? It’s mainly due to the following Theorem Let A be a Hopf algebra. Then the category of A-modules (=representations of A) is braided(=has intertwiners) if and only if A is quasitriangular. What quantum groups have to do with QYBE? Proposition Let (A, R) be a quasitriangular Hopf algebra. Then the following form of QYBE holds in A⊗3 : R12 R13 R23 = R23 R13 R12 An upshot: finite-dimensional representations of a quantum group give rise to an R-matrices. For this reason, R is sometimes called a universal R-matrix. UMN Math Physics Seminar Quantum groups and integrable lattice models Hopf algebras and quantum groups What is the meaning of the quasitriangular condition? It’s mainly due to the following Theorem Let A be a Hopf algebra. Then the category of A-modules (=representations of A) is braided(=has intertwiners) if and only if A is quasitriangular. What quantum groups have to do with QYBE? Proposition Let (A, R) be a quasitriangular Hopf algebra. Then the following form of QYBE holds in A⊗3 : R12 R13 R23 = R23 R13 R12 An upshot: finite-dimensional representations of a quantum group give rise to an R-matrices. For this reason, R is sometimes called a universal R-matrix. UMN Math Physics Seminar Quantum groups and integrable lattice models Quantum groups from affine Lie algebras A rich source of quantum groups comes from affine Lie algebras. Start with a simple finite-dimensional Lie algebra g. Consider a central extension ĝ of its loop algebra g ⊗ C[t, t−1 ]. There is a way to deform the universal enveloping algebra of ĝ into the so-called quantum UEA Uq (ĝ), which has a structure of a quantum group. Under favorable conditions, one can obtain a family of Uq (ĝ)-modules Vζ , ζ ∈ C and the universal R-matrix of Uq (ĝ) would give rise to the intertwining operators R(ζ1 , ζ2 ) : Vζ1 ⊗ Vζ2 → Vζ2 ⊗ Vζ1 . These intertwiners satify the QYBE. Thus we got a family of R-matrices. UMN Math Physics Seminar Quantum groups and integrable lattice models Quantum groups from affine Lie algebras A rich source of quantum groups comes from affine Lie algebras. Start with a simple finite-dimensional Lie algebra g. Consider a central extension ĝ of its loop algebra g ⊗ C[t, t−1 ]. There is a way to deform the universal enveloping algebra of ĝ into the so-called quantum UEA Uq (ĝ), which has a structure of a quantum group. Under favorable conditions, one can obtain a family of Uq (ĝ)-modules Vζ , ζ ∈ C and the universal R-matrix of Uq (ĝ) would give rise to the intertwining operators R(ζ1 , ζ2 ) : Vζ1 ⊗ Vζ2 → Vζ2 ⊗ Vζ1 . These intertwiners satify the QYBE. Thus we got a family of R-matrices. UMN Math Physics Seminar Quantum groups and integrable lattice models Quantum groups from affine Lie algebras A rich source of quantum groups comes from affine Lie algebras. Start with a simple finite-dimensional Lie algebra g. Consider a central extension ĝ of its loop algebra g ⊗ C[t, t−1 ]. There is a way to deform the universal enveloping algebra of ĝ into the so-called quantum UEA Uq (ĝ), which has a structure of a quantum group. Under favorable conditions, one can obtain a family of Uq (ĝ)-modules Vζ , ζ ∈ C and the universal R-matrix of Uq (ĝ) would give rise to the intertwining operators R(ζ1 , ζ2 ) : Vζ1 ⊗ Vζ2 → Vζ2 ⊗ Vζ1 . These intertwiners satify the QYBE. Thus we got a family of R-matrices. UMN Math Physics Seminar Quantum groups and integrable lattice models Quantum groups from affine Lie algebras A rich source of quantum groups comes from affine Lie algebras. Start with a simple finite-dimensional Lie algebra g. Consider a central extension ĝ of its loop algebra g ⊗ C[t, t−1 ]. There is a way to deform the universal enveloping algebra of ĝ into the so-called quantum UEA Uq (ĝ), which has a structure of a quantum group. Under favorable conditions, one can obtain a family of Uq (ĝ)-modules Vζ , ζ ∈ C and the universal R-matrix of Uq (ĝ) would give rise to the intertwining operators R(ζ1 , ζ2 ) : Vζ1 ⊗ Vζ2 → Vζ2 ⊗ Vζ1 . These intertwiners satify the QYBE. Thus we got a family of R-matrices. UMN Math Physics Seminar Quantum groups and integrable lattice models Quantum groups from affine Lie algebras A rich source of quantum groups comes from affine Lie algebras. Start with a simple finite-dimensional Lie algebra g. Consider a central extension ĝ of its loop algebra g ⊗ C[t, t−1 ]. There is a way to deform the universal enveloping algebra of ĝ into the so-called quantum UEA Uq (ĝ), which has a structure of a quantum group. Under favorable conditions, one can obtain a family of Uq (ĝ)-modules Vζ , ζ ∈ C and the universal R-matrix of Uq (ĝ) would give rise to the intertwining operators R(ζ1 , ζ2 ) : Vζ1 ⊗ Vζ2 → Vζ2 ⊗ Vζ1 . These intertwiners satify the QYBE. Thus we got a family of R-matrices. UMN Math Physics Seminar Quantum groups and integrable lattice models Quantum groups from affine Lie algebras Example The above construction applied to sl2 produces R-matrices of the six-vertex model. UMN Math Physics Seminar Quantum groups and integrable lattice models