Quantum Toroidal Superalgebras
Luan BezerraJoint work with Evgeny Mukhin
IUPUI
2019
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 1 / 14
Quantum Toroidal Superalgebra Em|n(q1, q2, q3).
1 Definition.
2 Vertex Representations.
3 Evaluation Homomorphism.
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 2 / 14
Quantum Toroidal Superalgebra Em|n(q1, q2, q3).
1 Definition.
2 Vertex Representations.
3 Evaluation Homomorphism.
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 3 / 14
Quantum Toroidal Superalgebra Em|n(q1, q2, q3).
1 Definition.
2 Vertex Representations.
3 Evaluation Homomorphism.
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 4 / 14
Quantum Toroidal Superalgebra Em|n(q1, q2, q3).
It is an affinization of Uq glm|n with m, n ≥ 1 and m 6= n;
Generators: Ei (z),Fi (z),K±i (z),C , i ∈ I ;
m + n − 1 m + 2 m + 1
m − 121
0 m
Depends on two complex parameters: q1q2q3 = 1, q2 = q2;
It has a vertical subalgebra Uverq slm|n ∼= Uq slm|n given in new Drinfeld
realization,
and a horizontal subalgebra Uhorq slm|n ∼= Uq slm|n given in
Drinfeld-Jimbo realization.
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 5 / 14
Quantum Toroidal Superalgebra Em|n(q1, q2, q3).
It is an affinization of Uq glm|n with m, n ≥ 1 and m 6= n;
Generators: Ei (z),Fi (z),K±i (z),C , i ∈ I ;
m + n − 1 m + 2 m + 1
m − 121
0 m
Depends on two complex parameters: q1q2q3 = 1, q2 = q2;
It has a vertical subalgebra Uverq slm|n ∼= Uq slm|n given in new Drinfeld
realization,
and a horizontal subalgebra Uhorq slm|n ∼= Uq slm|n given in
Drinfeld-Jimbo realization.
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 5 / 14
Quantum Toroidal Superalgebra Em|n(q1, q2, q3).
It is an affinization of Uq glm|n with m, n ≥ 1 and m 6= n;
Generators: Ei (z),Fi (z),K±i (z),C , i ∈ I ;
m + n − 1 m + 2 m + 1
m − 121
0 m
Depends on two complex parameters: q1q2q3 = 1, q2 = q2;
It has a vertical subalgebra Uverq slm|n ∼= Uq slm|n given in new Drinfeld
realization,
and a horizontal subalgebra Uhorq slm|n ∼= Uq slm|n given in
Drinfeld-Jimbo realization.
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 5 / 14
Quantum Toroidal Superalgebra Em|n(q1, q2, q3).
It is an affinization of Uq glm|n with m, n ≥ 1 and m 6= n;
Generators: Ei (z),Fi (z),K±i (z),C , i ∈ I ;
m + n − 1 m + 2 m + 1
m − 121
0 m
Depends on two complex parameters: q1q2q3 = 1, q2 = q2;
It has a vertical subalgebra Uverq slm|n ∼= Uq slm|n given in new Drinfeld
realization,
and a horizontal subalgebra Uhorq slm|n ∼= Uq slm|n given in
Drinfeld-Jimbo realization.
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 5 / 14
Quantum Toroidal Superalgebra Em|n(q1, q2, q3).
It is an affinization of Uq glm|n with m, n ≥ 1 and m 6= n;
Generators: Ei (z),Fi (z),K±i (z),C , i ∈ I ;
m + n − 1 m + 2 m + 1
m − 121
0 m
Depends on two complex parameters: q1q2q3 = 1, q2 = q2;
It has a vertical subalgebra Uverq slm|n ∼= Uq slm|n given in new Drinfeld
realization,
and a horizontal subalgebra Uhorq slm|n ∼= Uq slm|n given in
Drinfeld-Jimbo realization.
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 5 / 14
Relations
KiKj = KjKi , KiEj(z)K−1i = qAi,jEj(z) , K±i (z)K±j (w) = K±j (w)K±i (z) ,
K±i (z)K±j (w) = K±j (w)K±i (z) ,
dMi,jC−1z − qAi,jw
dMi,jCz − qAi,jwK−i (z)K+
j (w) =dMi,jqAi,jC−1z − w
dMi,jqAi,jCz − wK+j (w)K−i (z) ,
(dMi,j z − qAi,jw)K±i (C−1±12 z)Ej(w) = (dMi,jqAi,j z − w)Ej(w)K±i (C−
1±12 z) ,
[Ei (z),Fj(w)] =δi ,j
q − q−1(δ(Cw
z
)K+i (w)− δ
(C
z
w
)K−i (z)) ,
[Ei (z),Ej(w)] = 0 , [Fi (z),Fj(w)] = 0 (Ai,j=0),
(dMi,j z − qAi,jw)Ei (z)Ej(w) = (−1)|i ||j |(dMi,jqAi,j z − w)Ej(w)Ei (z) (Ai,j 6=0),
+ Serre relations.
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 6 / 14
Properties
Diagram isomorphism
σ : Em|n(q1, q2, q3)→ Em|n(q3, q2, q1), σ(Ei (z)) = Em−i (z);
change of parity isomorphism
τ : Em|n(q1, q2, q3)→ En|m(q−13 , q−12 , q−11 ), τ(Ei (z)) = E−i (z);
it has a topological Hopf superalgebra structure:
∆Ei (z) = Ei (z)⊗ 1 + K−i (z)⊗ Ei (C ⊗ z);
it has a two dimensional center.
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 7 / 14
Properties
Diagram isomorphism
σ : Em|n(q1, q2, q3)→ Em|n(q3, q2, q1), σ(Ei (z)) = Em−i (z);
change of parity isomorphism
τ : Em|n(q1, q2, q3)→ En|m(q−13 , q−12 , q−11 ), τ(Ei (z)) = E−i (z);
it has a topological Hopf superalgebra structure:
∆Ei (z) = Ei (z)⊗ 1 + K−i (z)⊗ Ei (C ⊗ z);
it has a two dimensional center.
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 7 / 14
Properties
Diagram isomorphism
σ : Em|n(q1, q2, q3)→ Em|n(q3, q2, q1), σ(Ei (z)) = Em−i (z);
change of parity isomorphism
τ : Em|n(q1, q2, q3)→ En|m(q−13 , q−12 , q−11 ), τ(Ei (z)) = E−i (z);
it has a topological Hopf superalgebra structure:
∆Ei (z) = Ei (z)⊗ 1 + K−i (z)⊗ Ei (C ⊗ z);
it has a two dimensional center.
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 7 / 14
Properties
Diagram isomorphism
σ : Em|n(q1, q2, q3)→ Em|n(q3, q2, q1), σ(Ei (z)) = Em−i (z);
change of parity isomorphism
τ : Em|n(q1, q2, q3)→ En|m(q−13 , q−12 , q−11 ), τ(Ei (z)) = E−i (z);
it has a topological Hopf superalgebra structure:
∆Ei (z) = Ei (z)⊗ 1 + K−i (z)⊗ Ei (C ⊗ z);
it has a two dimensional center.
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 7 / 14
Properties
Different parity choices yield isomorphic superalgebras;
Miki automorphism: interchanges the vertical and horizontalsubalgebras.
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 8 / 14
Properties
Different parity choices yield isomorphic superalgebras;
Miki automorphism: interchanges the vertical and horizontalsubalgebras.
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 8 / 14
Level 1 Modules - Vertex Operators
The Frenkel-Kac construction of Uq gln-modules was extended to thequantum toroidal (purely even) case [S]. We use the same techniqueto extend the construction of level 1 Uq glm|n-modules [KSU] toEm|n-modules.
As in the affine case, the modules obtained are not irreducible.
It is conjectured the irreducible modules are the kernel (or cokernel)of some screening operators.
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 9 / 14
Level 1 Modules - Vertex Operators
The Frenkel-Kac construction of Uq gln-modules was extended to thequantum toroidal (purely even) case [S]. We use the same techniqueto extend the construction of level 1 Uq glm|n-modules [KSU] toEm|n-modules.
As in the affine case, the modules obtained are not irreducible.
It is conjectured the irreducible modules are the kernel (or cokernel)of some screening operators.
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 9 / 14
Level 1 Modules - Vertex Operators
The Frenkel-Kac construction of Uq gln-modules was extended to thequantum toroidal (purely even) case [S]. We use the same techniqueto extend the construction of level 1 Uq glm|n-modules [KSU] toEm|n-modules.
As in the affine case, the modules obtained are not irreducible.
It is conjectured the irreducible modules are the kernel (or cokernel)of some screening operators.
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 9 / 14
Evaluation Map
Theorem
Fix u ∈ C×. We have a surjective homomorphism of superalgebrasevu : Em|n → Uq glm|n with c2 = qm−n3 . In particular, any admissible
Uqglm|n-module on which the central element c acts as an arbitrary scalar
α can be lifted to an Em|n-module choosing q3 satisfying α2 = qm−n3 .
Generators with index i ∈ I (Uverq slm|n) are mapped to generators;
Generators corresponding to the node 0 are mapped to “dressed”currents:
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 10 / 14
Evaluation Map
Theorem
Fix u ∈ C×. We have a surjective homomorphism of superalgebrasevu : Em|n → Uq glm|n with c2 = qm−n3 . In particular, any admissible
Uqglm|n-module on which the central element c acts as an arbitrary scalar
α can be lifted to an Em|n-module choosing q3 satisfying α2 = qm−n3 .
Generators with index i ∈ I (Uverq slm|n) are mapped to generators;
Generators corresponding to the node 0 are mapped to “dressed”currents:
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 10 / 14
Evaluation Map
Theorem
Fix u ∈ C×. We have a surjective homomorphism of superalgebrasevu : Em|n → Uq glm|n with c2 = qm−n3 . In particular, any admissible
Uqglm|n-module on which the central element c acts as an arbitrary scalar
α can be lifted to an Em|n-module choosing q3 satisfying α2 = qm−n3 .
Generators with index i ∈ I (Uverq slm|n) are mapped to generators;
Generators corresponding to the node 0 are mapped to “dressed”currents:
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 10 / 14
Evaluation Map
E0(z) 7→ u−1
exp(A−(z)
)X−(z)exp
(A+(z)
)K
,
K is in the weight lattice of Uqglm|n,
A±(z) =∑
r>0 A±rz∓r ,
Ar = − q−q−1
c r−c−r
(h0,r +
∑mi=1(c2qi )rhi ,r +
∑m+n−1j=m+1 (c2q2m−j)rhj ,r
),
X−(z) =
[∏m+n−2i=1
(1− zi+1
zi
)]x−1 (q−1c−1z1) · · · x−m (q−mc−1zm)×
× · · · x−m+i (q−m+ic−1zm+i ) · · · x−m+n−1(q−m+n−1c−1zm+n−1)
∣∣∣z1=···=zm+n−1=z
,
x−i (z) are current generators of Uqglm|n.
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 11 / 14
Evaluation Map
E0(z) 7→ u−1
exp(A−(z)
)
X−(z)
exp(A+(z)
)K
,
K is in the weight lattice of Uqglm|n,
A±(z) =∑
r>0 A±rz∓r ,
Ar = − q−q−1
c r−c−r
(h0,r +
∑mi=1(c2qi )rhi ,r +
∑m+n−1j=m+1 (c2q2m−j)rhj ,r
),
X−(z) =
[∏m+n−2i=1
(1− zi+1
zi
)]x−1 (q−1c−1z1) · · · x−m (q−mc−1zm)×
× · · · x−m+i (q−m+ic−1zm+i ) · · · x−m+n−1(q−m+n−1c−1zm+n−1)
∣∣∣z1=···=zm+n−1=z
,
x−i (z) are current generators of Uqglm|n.
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 11 / 14
Evaluation Map
E0(z) 7→ u−1
exp(A−(z)
)
X−(z)
exp(A+(z)
)
K,
K is in the weight lattice of Uqglm|n,
A±(z) =∑
r>0 A±rz∓r ,
Ar = − q−q−1
c r−c−r
(h0,r +
∑mi=1(c2qi )rhi ,r +
∑m+n−1j=m+1 (c2q2m−j)rhj ,r
),
X−(z) =
[∏m+n−2i=1
(1− zi+1
zi
)]x−1 (q−1c−1z1) · · · x−m (q−mc−1zm)×
× · · · x−m+i (q−m+ic−1zm+i ) · · · x−m+n−1(q−m+n−1c−1zm+n−1)
∣∣∣z1=···=zm+n−1=z
,
x−i (z) are current generators of Uqglm|n.
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 11 / 14
Evaluation Map
E0(z) 7→ u−1exp(A−(z)
)X−(z)exp
(A+(z)
)K,
K is in the weight lattice of Uqglm|n,
A±(z) =∑
r>0 A±rz∓r ,
Ar = − q−q−1
c r−c−r
(h0,r +
∑mi=1(c2qi )rhi ,r +
∑m+n−1j=m+1 (c2q2m−j)rhj ,r
),
X−(z) =
[∏m+n−2i=1
(1− zi+1
zi
)]x−1 (q−1c−1z1) · · · x−m (q−mc−1zm)×
× · · · x−m+i (q−m+ic−1zm+i ) · · · x−m+n−1(q−m+n−1c−1zm+n−1)
∣∣∣z1=···=zm+n−1=z
,
x−i (z) are current generators of Uqglm|n.
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 11 / 14
Evaluation Map
E0(z) 7→ u−1exp(A−(z)
)X−(z)exp
(A+(z)
)K,
K is in the weight lattice of Uqglm|n,
A±(z) =∑
r>0 A±rz∓r ,
Ar = − q−q−1
c r−c−r
(h0,r +
∑mi=1(c2qi )rhi ,r +
∑m+n−1j=m+1 (c2q2m−j)rhj ,r
),
X−(z) =
[∏m+n−2i=1
(1− zi+1
zi
)]x−1 (q−1c−1z1) · · · x−m (q−mc−1zm)×
× · · · x−m+i (q−m+ic−1zm+i ) · · · x−m+n−1(q−m+n−1c−1zm+n−1)
∣∣∣z1=···=zm+n−1=z
,
x−i (z) are current generators of Uqglm|n.
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 11 / 14
References
K. Kimura, J. Shiraishi, and J. Uchiyama, A level-one representation
of the quantum affine superalgebra Uq(sl(M + 1|N + 1)), Comm.Math. Phys. 188 (1997), no. 2, 367—378
K. Miki, Toroidal braid group action and an automorphism of toroidalalgebra Uq
(sln+1,tor
)(n ≥ 2), Lett. Math. Phys. 47 (1999), no. 4,
365–378
Y. Saito, Quantum toroidal algebras and their vertex representations,Publ. Res. Inst. Math. Sci. 34 (1998), no. 2, 155—177
H. Yamane, On defining relations of affine Lie superalgebras and affinequantized universal enveloping superalgebras, Publ. RIMS, KyotoUniv. 35 (1999), 321-–390
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 12 / 14
Thank you!
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 13 / 14
Happy Birthday,Prof. Tarasov and Prof. Varchenko!
Luan Bezerra (IUPUI) Quantum Toroidal Superalgebras 2019 14 / 14