Quantum Toroidal Superalgebrasrtis2019.math.iupui.edu/wp-content/uploads/2019/08/Bezerra.pdf ·...

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