Download Quantum groups and integrable lattice models UMN Math Physics Seminar

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