From quantum Q system to quantum toroidal algebras via Macdonald - - PowerPoint PPT Presentation

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

From quantum Q system to quantum toroidal algebras via Macdonald difference operators

Rinat Kedem (Joint work with Philippe Di Francesco)

University of Illinois

Rome 2016

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Outline

1 Quantum toroidal algebra 2 Generalized Macdonald difference operators 3 q-Whittaker limit 4 The quantum Q-system and quantum affine algebra 5 Conclusion

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Current algebras

The Lie algebra sl2 has generators e, f , h with relations. [e, f ] = h; [h, e] = 2e; [h, f ] = −2f . The Lie algebra b sl2 ≃ sl2 ⊗ C[u, u−1] ⊕ Cc has generators {e[n], f [n], h[n] : n ∈ Z}. Relations expressed using generating functions (currents): e(z) = X

n∈Z

e[n]zn, f (z) = X

n∈Z

f [n]zn, h(z) = X

n∈Z

h[n]zn. The key function when working with generating functions: The delta function δ(z) = X

n

zn = 1 1 − z + z−1 1 − z−1 .

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Current algebras

The Lie algebra sl2 has generators e, f , h with relations. [e, f ] = h; [h, e] = 2e; [h, f ] = −2f . The Lie algebra b sl2 ≃ sl2 ⊗ C[u, u−1] ⊕ Cc has generators {e[n], f [n], h[n] : n ∈ Z}. Relations expressed using generating functions (currents): e(z) = X

n∈Z

e[n]zn, f (z) = X

n∈Z

f [n]zn, h(z) = X

n∈Z

h[n]zn. The key function when working with generating functions: The delta function δ(z) = X

n

zn = 1 1 − z + z−1 1 − z−1 .

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Current algebras

The Lie algebra sl2 has generators e, f , h with relations. [e, f ] = h; [h, e] = 2e; [h, f ] = −2f . The Lie algebra b sl2 ≃ sl2 ⊗ C[u, u−1] ⊕ Cc has generators {e[n], f [n], h[n] : n ∈ Z}. Relations expressed using generating functions (currents): e(z) = X

n∈Z

e[n]zn, f (z) = X

n∈Z

f [n]zn, h(z) = X

n∈Z

h[n]zn. The key function when working with generating functions: The delta function δ(z) = X

n

zn = 1 1 − z + z−1 1 − z−1 .

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Current algebras

The Lie algebra sl2 has generators e, f , h with relations. [e, f ] = h; [h, e] = 2e; [h, f ] = −2f . The Lie algebra b sl2 ≃ sl2 ⊗ C[u, u−1] ⊕ Cc has generators {e[n], f [n], h[n] : n ∈ Z}. Relations expressed using generating functions (currents): e(z) = X

n∈Z

e[n]zn, f (z) = X

n∈Z

f [n]zn, h(z) = X

n∈Z

h[n]zn. The key function when working with generating functions: The delta function δ(z) = X

n

zn = 1 1 − z + z−1 1 − z−1 .

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Drinfeld-type relations

The relations for affine sl2 in terms of generating functions: [h(z), e(w)] = 2δ(z/w)e(z), [h(z), f (w)] = −2δ(z/w)f (z), [e(z), f (w)] = δ(z/w)h(z) + δ′(z/w)c. But [e(z), e(w)] = 0, [f (z), f (w)] = 0, [h(z), h(w)] = 0. and c is the central element. The current relations are shorthand for relations among (infinitely many)

• generators. For example, the third relation in the algebra is

[e[n], f [m]] = h[n + m] + nδn,−mc.

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Drinfeld-type relations

The relations for affine sl2 in terms of generating functions: [h(z), e(w)] = 2δ(z/w)e(z), [h(z), f (w)] = −2δ(z/w)f (z), [e(z), f (w)] = δ(z/w)h(z) + δ′(z/w)c. But [e(z), e(w)] = 0, [f (z), f (w)] = 0, [h(z), h(w)] = 0. and c is the central element. The current relations are shorthand for relations among (infinitely many)

• generators. For example, the third relation in the algebra is

[e[n], f [m]] = h[n + m] + nδn,−mc.

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Ding-Iohara deformations of U( sl2)

Fix a function g ∈ F(z, w), and define the universal enveloping algebra U = Ug using current generators E(z), F(z) ∈ U[[z, z−1]] and Ψ±(z) ∈ U[[z±1]]. In this talk, take the central element to be C = 0. The non-trivial relations are g(z, w)E(z)E(w) + E(w)E(z)g(w, z) = 0 g(w, z)F(z)F(w) + F(w)F(z)g(z, w) = 0 g(z, w)Ψ±(z)E(w) + g(w, z)E(w)Ψ±(z) = 0 g(w, z)Ψ±(z)F(w) + g(z, w)F(w)Ψ±(z) = 0 [E(z), F(w)] = 1 g(1, 1)δ(z/w) ` Ψ+(z) − Ψ−(w) ´ + possibly Serre-type (cubic) relations.

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Ding-Iohara deformations of U( sl2)

Fix a function g ∈ F(z, w), and define the universal enveloping algebra U = Ug using current generators E(z), F(z) ∈ U[[z, z−1]] and Ψ±(z) ∈ U[[z±1]]. In this talk, take the central element to be C = 0. The non-trivial relations are g(z, w)E(z)E(w) + E(w)E(z)g(w, z) = 0 g(w, z)F(z)F(w) + F(w)F(z)g(z, w) = 0 g(z, w)Ψ±(z)E(w) + g(w, z)E(w)Ψ±(z) = 0 g(w, z)Ψ±(z)F(w) + g(z, w)F(w)Ψ±(z) = 0 [E(z), F(w)] = 1 g(1, 1)δ(z/w) ` Ψ+(z) − Ψ−(w) ´ + possibly Serre-type (cubic) relations.

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Two examples of g(z, w)

g(z, w)E(z)E(w) + E(w)E(z)g(w, z) = 0 g(w, z)F(z)F(w) + F(w)F(z)g(z, w) = 0 g(z, w)Ψ±(z)E(w) + g(w, z)E(w)Ψ±(z) = 0 g(w, z)Ψ±(z)F(w) + g(z, w)F(w)Ψ±(z) = 0 [E(z), F(w)] = 1 g(1, 1)δ(z/w) ` Ψ+(z) − Ψ−(w) ´ When g(z) = (z − q2w), the relations define the algebra Uq(b sl 2) at zero central charge. When g(z) = (z − qw)(z − t−1w)(z − q−1tw) the algebra is the “Ding-Iohara” algebra. If we add a “Serre relation” Symzi „z2 z3 [E(z1), [E(z2), E(z3)]] « = 0, and similarly for F(z), the algebra is the “quantum toroidal algebra” [Miki, Feigin et al].

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Two examples of g(z, w)

g(z, w)E(z)E(w) + E(w)E(z)g(w, z) = 0 g(w, z)F(z)F(w) + F(w)F(z)g(z, w) = 0 g(z, w)Ψ±(z)E(w) + g(w, z)E(w)Ψ±(z) = 0 g(w, z)Ψ±(z)F(w) + g(z, w)F(w)Ψ±(z) = 0 [E(z), F(w)] = 1 g(1, 1)δ(z/w) ` Ψ+(z) − Ψ−(w) ´ When g(z) = (z − q2w), the relations define the algebra Uq(b sl 2) at zero central charge. When g(z) = (z − qw)(z − t−1w)(z − q−1tw) the algebra is the “Ding-Iohara” algebra. If we add a “Serre relation” Symzi „z2 z3 [E(z1), [E(z2), E(z3)]] « = 0, and similarly for F(z), the algebra is the “quantum toroidal algebra” [Miki, Feigin et al].

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Two examples of g(z, w)

g(z, w)E(z)E(w) + E(w)E(z)g(w, z) = 0 g(w, z)F(z)F(w) + F(w)F(z)g(z, w) = 0 g(z, w)Ψ±(z)E(w) + g(w, z)E(w)Ψ±(z) = 0 g(w, z)Ψ±(z)F(w) + g(z, w)F(w)Ψ±(z) = 0 [E(z), F(w)] = 1 g(1, 1)δ(z/w) ` Ψ+(z) − Ψ−(w) ´ When g(z) = (z − q2w), the relations define the algebra Uq(b sl 2) at zero central charge. When g(z) = (z − qw)(z − t−1w)(z − q−1tw) the algebra is the “Ding-Iohara” algebra. If we add a “Serre relation” Symzi „z2 z3 [E(z1), [E(z2), E(z3)]] « = 0, and similarly for F(z), the algebra is the “quantum toroidal algebra” [Miki, Feigin et al].

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Partial dictionary of algebras

The quantum toroidal algebra Utor(b gl 1) was introduced by Miki (2000 and 2007). Also known as the deformed W1+∞-algebra. Schiffmann proved the equivalence to the elliptic Hall algebra. Schiffmann-Vasserot gave an isomorphism: Elliptic Hall ≃ spherical DAHA

• f type AN when N → ∞.

The limit t → ∞ of the quantum toroidal algebra is a quantum affine algebra (after renormalization of generators). U√q(b sl2) or half of it: U√q(n+[u, u−1]). [Hall algebra, Kapranov].

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Representation by difference operators

There is a representation of the quantum toroidal algebra at zero central charge as difference operators acting on the space of symmetric Laurent polynomials in infinitely many variables. The Macdonald operators Dα(x1, ..., xN), 1 ≤ α ≤ N are Dα(x1, ..., xN) = X

I:|I|=α

Y

i∈I,j / ∈I

t

1 2 xi − t− 1 2 xj

xi − xj DI, where DI acts by multiplication of xi by q for all i ∈ I.

1 Preserve the space of symmetric polynomials in N variables with coefficients

in C(q, t);

2 Commute with each other: [Dα, Dβ] = 0. 3 Eigenvectors in the space of symmetric polynomials are the Macdonald

polynomials.

4 Make sense in the limit N → ∞.

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Representation by difference operators

There is a representation of the quantum toroidal algebra at zero central charge as difference operators acting on the space of symmetric Laurent polynomials in infinitely many variables. The Macdonald operators Dα(x1, ..., xN), 1 ≤ α ≤ N are Dα(x1, ..., xN) = X

I:|I|=α

Y

i∈I,j / ∈I

t

1 2 xi − t− 1 2 xj

xi − xj DI, where DI acts by multiplication of xi by q for all i ∈ I.

1 Preserve the space of symmetric polynomials in N variables with coefficients

in C(q, t);

2 Commute with each other: [Dα, Dβ] = 0. 3 Eigenvectors in the space of symmetric polynomials are the Macdonald

polynomials.

4 Make sense in the limit N → ∞.

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Representation by difference operators

There is a representation of the quantum toroidal algebra at zero central charge as difference operators acting on the space of symmetric Laurent polynomials in infinitely many variables. The Macdonald operators Dα(x1, ..., xN), 1 ≤ α ≤ N are Dα(x1, ..., xN) = X

I:|I|=α

Y

i∈I,j / ∈I

t

1 2 xi − t− 1 2 xj

xi − xj DI, where DI acts by multiplication of xi by q for all i ∈ I.

1 Preserve the space of symmetric polynomials in N variables with coefficients

in C(q, t);

2 Commute with each other: [Dα, Dβ] = 0. 3 Eigenvectors in the space of symmetric polynomials are the Macdonald

polynomials.

4 Make sense in the limit N → ∞.

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Representation by difference operators

There is a representation of the quantum toroidal algebra at zero central charge as difference operators acting on the space of symmetric Laurent polynomials in infinitely many variables. The Macdonald operators Dα(x1, ..., xN), 1 ≤ α ≤ N are Dα(x1, ..., xN) = X

I:|I|=α

Y

i∈I,j / ∈I

t

1 2 xi − t− 1 2 xj

xi − xj DI, where DI acts by multiplication of xi by q for all i ∈ I.

1 Preserve the space of symmetric polynomials in N variables with coefficients

in C(q, t);

2 Commute with each other: [Dα, Dβ] = 0. 3 Eigenvectors in the space of symmetric polynomials are the Macdonald

polynomials.

4 Make sense in the limit N → ∞.

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Representation by difference operators

There is a representation of the quantum toroidal algebra at zero central charge as difference operators acting on the space of symmetric Laurent polynomials in infinitely many variables. The Macdonald operators Dα(x1, ..., xN), 1 ≤ α ≤ N are Dα(x1, ..., xN) = X

I:|I|=α

Y

i∈I,j / ∈I

t

1 2 xi − t− 1 2 xj

xi − xj DI, where DI acts by multiplication of xi by q for all i ∈ I.

1 Preserve the space of symmetric polynomials in N variables with coefficients

in C(q, t);

2 Commute with each other: [Dα, Dβ] = 0. 3 Eigenvectors in the space of symmetric polynomials are the Macdonald

polynomials.

4 Make sense in the limit N → ∞.

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Representation by difference operators

There is a representation of the quantum toroidal algebra at zero central charge as difference operators acting on the space of symmetric Laurent polynomials in infinitely many variables. The Macdonald operators Dα(x1, ..., xN), 1 ≤ α ≤ N are Dα(x1, ..., xN) = X

I:|I|=α

Y

i∈I,j / ∈I

t

1 2 xi − t− 1 2 xj

xi − xj DI, where DI acts by multiplication of xi by q for all i ∈ I.

1 Preserve the space of symmetric polynomials in N variables with coefficients

in C(q, t);

2 Commute with each other: [Dα, Dβ] = 0. 3 Eigenvectors in the space of symmetric polynomials are the Macdonald

polynomials.

4 Make sense in the limit N → ∞.

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Generalized Macdonald operators

Define the generalization of the Macdonald operators: Dα;n(x1, ..., xN) = X

I:|I|=α

xn

I

Y

i∈I;j / ∈I

t

1 2 xi − t− 1 2 xj

xi − xj DI. where xI = Y

i∈I

xi. Theorem: For any fixed n, {Dα,n(x1, ..., xN) : α = 1, ..., N} is a commuting set of operators, preserve the space of symmetric Laurent polynomials. Remark: See Miki (2007) for α = 1 version.

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Generalized Macdonald operators

Define the generalization of the Macdonald operators: Dα;n(x1, ..., xN) = X

I:|I|=α

xn

I

Y

i∈I;j / ∈I

t

1 2 xi − t− 1 2 xj

xi − xj DI. where xI = Y

i∈I

xi. Theorem: For any fixed n, {Dα,n(x1, ..., xN) : α = 1, ..., N} is a commuting set of operators, preserve the space of symmetric Laurent polynomials. Remark: See Miki (2007) for α = 1 version.

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Generalized Macdonald operators

Define the generalization of the Macdonald operators: Dα;n(x1, ..., xN) = X

I:|I|=α

xn

I

Y

i∈I;j / ∈I

t

1 2 xi − t− 1 2 xj

xi − xj DI. where xI = Y

i∈I

xi. Theorem: For any fixed n, {Dα,n(x1, ..., xN) : α = 1, ..., N} is a commuting set of operators, preserve the space of symmetric Laurent polynomials. Remark: See Miki (2007) for α = 1 version.

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

The universal enveloping algebra (N → ∞)

What is the algebra generated by the generalized Macdonald operators? Claim: For any α > 1, Dα,m is in the universal enveloping algebra generated by {D1,n(n ∈ Z)}. That is, Dα,m is a polynomial in D1,n. Therefore {D1,n : n ∈ Z} are generators for the universal enveloping algebra U({Dα,m}). Define the generating function E(z) = 1 q− 1

2 − q 1 2

X

n∈Z

qn/2znD1,n = 1 q− 1

2 − q 1 2

X

i

δ(q

1 2 zxi)

Y

j=i

t

1 2 xi − t− 1 2 xj

xi − xj Di.

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

The universal enveloping algebra (N → ∞)

What is the algebra generated by the generalized Macdonald operators? Claim: For any α > 1, Dα,m is in the universal enveloping algebra generated by {D1,n(n ∈ Z)}. That is, Dα,m is a polynomial in D1,n. Therefore {D1,n : n ∈ Z} are generators for the universal enveloping algebra U({Dα,m}). Define the generating function E(z) = 1 q− 1

2 − q 1 2

X

n∈Z

qn/2znD1,n = 1 q− 1

2 − q 1 2

X

i

δ(q

1 2 zxi)

Y

j=i

t

1 2 xi − t− 1 2 xj

xi − xj Di.

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

The universal enveloping algebra (N → ∞)

What is the algebra generated by the generalized Macdonald operators? Claim: For any α > 1, Dα,m is in the universal enveloping algebra generated by {D1,n(n ∈ Z)}. That is, Dα,m is a polynomial in D1,n. Therefore {D1,n : n ∈ Z} are generators for the universal enveloping algebra U({Dα,m}). Define the generating function E(z) = 1 q− 1

2 − q 1 2

X

n∈Z

qn/2znD1,n = 1 q− 1

2 − q 1 2

X

i

δ(q

1 2 zxi)

Y

j=i

t

1 2 xi − t− 1 2 xj

xi − xj Di.

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Relation for the generating currents

Theorem: The generating function E(z) satisfies the following exchange relation: g(z, w)E(z)E(w) + E(w)E(z)g(w, z) = 0 with g(z, w) = (z − qw)(z − t−1w)(z − q−1tw) the function which defines the quantum toroidal algebra! Moreover, E(z) satisfies the “Serre relation”: Symzi „z2 z3 [E(z1), [E(z2), E(z3)]] « = 0. Relations in Utor(n+).

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Relation for the generating currents

Theorem: The generating function E(z) satisfies the following exchange relation: g(z, w)E(z)E(w) + E(w)E(z)g(w, z) = 0 with g(z, w) = (z − qw)(z − t−1w)(z − q−1tw) the function which defines the quantum toroidal algebra! Moreover, E(z) satisfies the “Serre relation”: Symzi „z2 z3 [E(z1), [E(z2), E(z3)]] « = 0. Relations in Utor(n+).

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

The rest of the algebra

The function g(z, w) has the symmetry: g(z, w) = (z − qw)(z − t−1w)(z − q−1tw) = −g(w, z) ˛ ˛ ˛

q→1/q,t→1/t.

Define the generating function F(z) = E(z) ˛ ˛ ˛

q→1/q,t→1/t :

F(z) = 1 q

1 2 − q− 1 2

X

i

δ(q− 1

2 zxi)

Y

j=i

t− 1

2 xi − t 1 2 xj

xi − xj D−1

i

. This generating function satisfies g(w, z)F(z)F(w) + g(z, w)F(w)F(z) = 0. plus Serre relations.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

The rest of the algebra

The function g(z, w) has the symmetry: g(z, w) = (z − qw)(z − t−1w)(z − q−1tw) = −g(w, z) ˛ ˛ ˛

q→1/q,t→1/t.

Define the generating function F(z) = E(z) ˛ ˛ ˛

q→1/q,t→1/t :

F(z) = 1 q

1 2 − q− 1 2

X

i

δ(q− 1

2 zxi)

Y

j=i

t− 1

2 xi − t 1 2 xj

xi − xj D−1

i

. This generating function satisfies g(w, z)F(z)F(w) + g(z, w)F(w)F(z) = 0. plus Serre relations.

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

The rest of the algebra

The function g(z, w) has the symmetry: g(z, w) = (z − qw)(z − t−1w)(z − q−1tw) = −g(w, z) ˛ ˛ ˛

q→1/q,t→1/t.

Define the generating function F(z) = E(z) ˛ ˛ ˛

q→1/q,t→1/t :

F(z) = 1 q

1 2 − q− 1 2

X

i

δ(q− 1

2 zxi)

Y

j=i

t− 1

2 xi − t 1 2 xj

xi − xj D−1

i

. This generating function satisfies g(w, z)F(z)F(w) + g(z, w)F(w)F(z) = 0. plus Serre relations.

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Cartan currents

The “Cartan currents” Ψ± can be computed from [E(z), F(w)] = 1 g(1, 1)δ(z/w)(Ψ+(z) − Ψ−(z)) Theorem: Ψ±(z) = Y

i

a((xiz)±1) where a(x) is a rational function of x. a(x) = (1 − q− 1

2 tx)(1 − q 1 2 t−1x)

(1 − q− 1

2 x)(1 − q 1 2 x)

. Note: Ψ±

0 = 1.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Cartan currents

The “Cartan currents” Ψ± can be computed from [E(z), F(w)] = 1 g(1, 1)δ(z/w)(Ψ+(z) − Ψ−(z)) Theorem: Ψ±(z) = Y

i

a((xiz)±1) where a(x) is a rational function of x. a(x) = (1 − q− 1

2 tx)(1 − q 1 2 t−1x)

(1 − q− 1

2 x)(1 − q 1 2 x)

. Note: Ψ±

0 = 1.

Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Cartan currents

The “Cartan currents” Ψ± can be computed from [E(z), F(w)] = 1 g(1, 1)δ(z/w)(Ψ+(z) − Ψ−(z)) Theorem: Ψ±(z) = Y

i

a((xiz)±1) where a(x) is a rational function of x. a(x) = (1 − q− 1

2 tx)(1 − q 1 2 t−1x)

(1 − q− 1

2 x)(1 − q 1 2 x)

. Note: Ψ±

0 = 1.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Other relations in the algebra

The cartan currents commute with E(z) and F(z) as g(z, w)Ψ±(z)E(w) + g(w, z)E(w)Ψ±(z) = 0, g(w, z)Ψ±(z)F(w) + g(z, w)F(w)Ψ±(z) = 0 The Serre relations are satisfied Symz2 z3 [E(z1), [E(z2), E(z3)]] = 0 Symz2 z3 [F(z1), [F(z2), F(z3)]] = 0 These relations are those of the quantum toroidal algebra (Miki, Feigin et al) Utor(gl1) at zero central charge.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Other relations in the algebra

The cartan currents commute with E(z) and F(z) as g(z, w)Ψ±(z)E(w) + g(w, z)E(w)Ψ±(z) = 0, g(w, z)Ψ±(z)F(w) + g(z, w)F(w)Ψ±(z) = 0 The Serre relations are satisfied Symz2 z3 [E(z1), [E(z2), E(z3)]] = 0 Symz2 z3 [F(z1), [F(z2), F(z3)]] = 0 These relations are those of the quantum toroidal algebra (Miki, Feigin et al) Utor(gl1) at zero central charge.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Other relations in the algebra

The cartan currents commute with E(z) and F(z) as g(z, w)Ψ±(z)E(w) + g(w, z)E(w)Ψ±(z) = 0, g(w, z)Ψ±(z)F(w) + g(z, w)F(w)Ψ±(z) = 0 The Serre relations are satisfied Symz2 z3 [E(z1), [E(z2), E(z3)]] = 0 Symz2 z3 [F(z1), [F(z2), F(z3)]] = 0 These relations are those of the quantum toroidal algebra (Miki, Feigin et al) Utor(gl1) at zero central charge.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Finite N

If there is only a finite number of variables xi on which the difference

• perators act, the relations in the algebra still hold, but there are extra

relations. The algebra of difference operators with a finite number of variables is a quotient of the quantum toroidal algebra. (See below). Also known as the deformed W1+∞ algebra. For finite number of variables, the algebra is isomorphic to the spherical DAHA of AN−1.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

The quantum limit t → ∞

The eigenfunctions of the Macdonald operators are Macdonald polynomials Pλ(x; q, t). They are symmetric under the exchange (q, t) → (q−1, t−1). When t → 0, the functions are known as spherical functions or q-Whittaker functions. The t → ∞ is the dual Whittaker limit.

q t t = q Hall Littlewood functions q-Whittaker functions S c h u r f u n c t i

• n

s Macdonald symmetric functions Pλ(z; q, t)

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Dual Whittaker limit of the algebra

When t → ∞, g(z, w) = (z − qw)(z − t−1w)(z − q−1tw) ∼ q−1twz(z − qw). The exchange relations for e(z) ∼ limt→∞ E(z) become: (z − qw)e(z)e(w) + (w − qz)e(w)e(z) = 0. That is a Drinfeld type relation with g(z, w) = (z − qw), a relation in U√q(b sl2). The relation for f (z) become (w − qz)f (z)f (w) + (z − qw)f (w)f (z) = 0.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

The Whittaker limit of the Cartan current

The only inhomogeneous relation is [E(z), F(w)] = δ(z/w) 1 g(1, 1)(Ψ+(z) − Ψ−(z)) But when t → ∞, 1/g(1, 1) = 1 (1 − q)(1 − t−1)(1 − tq−1) ∼ t−1 Choose some renormalization of E(z), F(z), Ψ±(z) so that the relations remain finite when t → ∞.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Difference operators at the dual Whittaker limit

Recall the difference operator presentation of the quantum toroidal algebra, E(z) = 1 q− 1

2 − q 1 2

X

i

δ(q

1 2 zxi)

Y

j=i

t

1 2 xi − t− 1 2 xj

xi − xj ∼ t(N−1)/2 where N is the number of variables. (If N finite, the relations are augmented by a relation accounting for finite N.) We define e(z) = lim

t→∞ t−(N−1)/2E(z),

similarly for f (z). Then ψ±(z) = lim

t→∞ t−NΨ±(z).

This normalization only makes sense for finite N.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Difference operators at t → ∞

The difference operators acting on the space of symmetric functions in N variables in the q-Whittaker limit: e(z) = 1 q− 1

2 − q 1 2

N

X

i=1

δ(q

1 2 zxi)

Y

j=i

xi xi − xj Di, f (z) = 1 q

1 2 − q− 1 2

N

X

i=1

δ(q− 1

2 zxi)

Y

j=i

xj xi − xj D−1

i

, and ψ±(z) = (−q− 1

2 z±1)N

N

Y

i=1

x±1

i

(1 − q− 1

2 (zxi)±1)(1 − q 1 2 (zxi)±1)

.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

The algebra of difference operators at t → ∞

When t → ∞ the difference operators e(z), f (z), ψ±(z) satisfy the current algebra relations with the function g(z, w) = (z − qw). However the Cartan currents have a non-standard valuation: ψ±(z) ∼ z±N + O(±zN+1). Not the usual choice for affine algebra representations! In the limit N → ∞, e(z) and f (z) commute, ψ±(z) → 0.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Relation to quantum Q-system

The quantum Q-system of type A with one coefficient: Commutation relation Mα,nMβ,n+p = qmin(α,β)pMβ,n+pMα,n, n ∈ Z, α ∈ [0, N] if |p| ≤ |α − β| + 1. Quadratic (mutation) relation: qαMα,n+1Mα,n−1 = M2

α,n − Mα+1,nMα−1,n,

Boundary conditions M0,n = 1, MN+1,n = 0.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Relation to quantum Q-system

The quantum Q-system of type A with one coefficient: Commutation relation Mα,nMβ,n+p = qmin(α,β)pMβ,n+pMα,n, n ∈ Z, α ∈ [0, N] if |p| ≤ |α − β| + 1. Quadratic (mutation) relation: qαMα,n+1Mα,n−1 = M2

α,n − Mα+1,nMα−1,n,

Boundary conditions M0,n = 1, MN+1,n = 0.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Relation to quantum Q-system

The quantum Q-system of type A with one coefficient: Commutation relation Mα,nMβ,n+p = qmin(α,β)pMβ,n+pMα,n, n ∈ Z, α ∈ [0, N] if |p| ≤ |α − β| + 1. Quadratic (mutation) relation: qαMα,n+1Mα,n−1 = M2

α,n − Mα+1,nMα−1,n,

Boundary conditions M0,n = 1, MN+1,n = 0.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Relation to quantum Q-system

The quantum Q-system of type A with one coefficient: Commutation relation Mα,nMβ,n+p = qmin(α,β)pMβ,n+pMα,n, n ∈ Z, α ∈ [0, N] if |p| ≤ |α − β| + 1. Quadratic (mutation) relation: qαMα,n+1Mα,n−1 = M2

α,n − Mα+1,nMα−1,n,

Boundary conditions M0,n = 1, MN+1,n = 0. This corresponds to the quiver (MN,n = An∆, ∆Mα,n = qαnMα,n∆):

(r−1,n) (r−1,n+1) (r,n+1)

A

(3,n) (3,n+1)

. . . . . .

n+1 n

(1,n) (1,n+1) (2,n) (2,n+1) (r,n) Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Generating set

Notice that the exchange relation gives (with α = 1 and M0,n = 1) M2,n = M2

1,n − qM1,n+1M1,n−1

That is, M2,n is a polynomial with coefficients in Z[q] in {M1,n+p : |p| ≤ 1}. This is a general fact: Theorem The generators Mα,n with α > 1 are polynomials in M1,n+p with |p| ≤ α − 1. That is, Mα,n ∈ U({M1,m}). Explict formula as a q-commutator: Mα,n ∝ [[· · · [M1,n−α+1, M1,n−α+3]q2, M1,n−α+5]q3, · · · , M1,n+α−1]qα.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Generating set

Notice that the exchange relation gives (with α = 1 and M0,n = 1) M2,n = M2

1,n − qM1,n+1M1,n−1

That is, M2,n is a polynomial with coefficients in Z[q] in {M1,n+p : |p| ≤ 1}. This is a general fact: Theorem The generators Mα,n with α > 1 are polynomials in M1,n+p with |p| ≤ α − 1. That is, Mα,n ∈ U({M1,m}). Explict formula as a q-commutator: Mα,n ∝ [[· · · [M1,n−α+1, M1,n−α+3]q2, M1,n−α+5]q3, · · · , M1,n+α−1]qα.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Generating set

Notice that the exchange relation gives (with α = 1 and M0,n = 1) M2,n = M2

1,n − qM1,n+1M1,n−1

That is, M2,n is a polynomial with coefficients in Z[q] in {M1,n+p : |p| ≤ 1}. This is a general fact: Theorem The generators Mα,n with α > 1 are polynomials in M1,n+p with |p| ≤ α − 1. That is, Mα,n ∈ U({M1,m}). Explict formula as a q-commutator: Mα,n ∝ [[· · · [M1,n−α+1, M1,n−α+3]q2, M1,n−α+5]q3, · · · , M1,n+α−1]qα.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Generating set

Notice that the exchange relation gives (with α = 1 and M0,n = 1) M2,n = M2

1,n − qM1,n+1M1,n−1

That is, M2,n is a polynomial with coefficients in Z[q] in {M1,n+p : |p| ≤ 1}. This is a general fact: Theorem The generators Mα,n with α > 1 are polynomials in M1,n+p with |p| ≤ α − 1. That is, Mα,n ∈ U({M1,m}). Explict formula as a q-commutator: Mα,n ∝ [[· · · [M1,n−α+1, M1,n−α+3]q2, M1,n−α+5]q3, · · · , M1,n+α−1]qα.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Current formulation

Everything looks nicer in terms of generating functions Mα(z) = X

n∈Z

Mα,nzn Theorem The iterated q-commutator relation is equivalent to Mα(z) = CTz1,z2,...,zα @ Y

1≤i<j≤α

(1 − qzj/zi)M1(z1) · · · M1(zα)δ(z1 · · · zα/z) 1 A . This is a “q-determinant” type formula. The boundary condition for finite N, MN+1(z) = 0 is a polynomial relation in {M1,n} of degree N + 1.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Exchange relations

Theorem: The quantum Q-system implies the following relations among the generating functions M(z) = M1(z): (z − qw)M(z)M(w) + (w − qz)M(w)M(z) = 0. In terms of components, this means (Mj := M1,j): [Mn, Mn+p]q + [Mn+p−1, Mn+1]q = 0, where [a, b]q = ab − qba. If p = 1 this is the commutation relation: [Mn, Mn+1]q + [Mn, Mn+1]q = 2(MnMn+1 − qMn+1Mn) = 0. If p = 2 this is the sum of two Q-system relations: [Mn, Mn+2]q + [Mn+1, Mn+1]q = (MnMn+2 − qM2

n+1 + M2,n+1) − (qMn+2Mn − M2 n+1 + M2,n+1) = 0.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Exchange relations

Theorem: The quantum Q-system implies the following relations among the generating functions M(z) = M1(z): (z − qw)M(z)M(w) + (w − qz)M(w)M(z) = 0. In terms of components, this means (Mj := M1,j): [Mn, Mn+p]q + [Mn+p−1, Mn+1]q = 0, where [a, b]q = ab − qba. If p = 1 this is the commutation relation: [Mn, Mn+1]q + [Mn, Mn+1]q = 2(MnMn+1 − qMn+1Mn) = 0. If p = 2 this is the sum of two Q-system relations: [Mn, Mn+2]q + [Mn+1, Mn+1]q = (MnMn+2 − qM2

n+1 + M2,n+1) − (qMn+2Mn − M2 n+1 + M2,n+1) = 0.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Exchange relations

Theorem: The quantum Q-system implies the following relations among the generating functions M(z) = M1(z): (z − qw)M(z)M(w) + (w − qz)M(w)M(z) = 0. In terms of components, this means (Mj := M1,j): [Mn, Mn+p]q + [Mn+p−1, Mn+1]q = 0, where [a, b]q = ab − qba. If p = 1 this is the commutation relation: [Mn, Mn+1]q + [Mn, Mn+1]q = 2(MnMn+1 − qMn+1Mn) = 0. If p = 2 this is the sum of two Q-system relations: [Mn, Mn+2]q + [Mn+1, Mn+1]q = (MnMn+2 − qM2

n+1 + M2,n+1) − (qMn+2Mn − M2 n+1 + M2,n+1) = 0.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Exchange relations

Theorem: The quantum Q-system implies the following relations among the generating functions M(z) = M1(z): (z − qw)M(z)M(w) + (w − qz)M(w)M(z) = 0. In terms of components, this means (Mj := M1,j): [Mn, Mn+p]q + [Mn+p−1, Mn+1]q = 0, where [a, b]q = ab − qba. If p = 1 this is the commutation relation: [Mn, Mn+1]q + [Mn, Mn+1]q = 2(MnMn+1 − qMn+1Mn) = 0. If p = 2 this is the sum of two Q-system relations: [Mn, Mn+2]q + [Mn+1, Mn+1]q = (MnMn+2 − qM2

n+1 + M2,n+1) − (qMn+2Mn − M2 n+1 + M2,n+1) = 0.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Q-system and quantum affine algebra

The relation in U(M(z)) is the same as the relation between the currents in U√q(n+[u, u−1]) ⊂ U√q(b sl2). But Warning there is an extra relation: MN+1(z) = 0. Theorem The universal enveloping algebra of the solutions of the slN Q-system is isomorphic to a quotient of (half of) the quantum affine algebra. In the limit N → ∞, the extra relation vanishes.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Integrable structure

In the cluster algebra formulation, we consider the localization of the algebra: We adjoin the inverses of the generators Mα,n to the algebra. Theorem: The elements in Uloc xα,n = Mα,n+1Mα−1,nM−1

α,nM−1 α,n+1 ∈ Uloc

are mutually commuting for fixed n, and have a well-defined limit as n → ∞: xα := limn→∞ xα,n. The Q-system is integrable: Theorem The “Miura operator” µn(z) = (D − zxN,n)(D − zxN−1,n) · · · (D − zx1,n) is independent of n: µn = µn+1. The proof is a “zipper proof”, using repeated application of the Q-system, starting from the right of the expression for µn.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Integrable structure

In the cluster algebra formulation, we consider the localization of the algebra: We adjoin the inverses of the generators Mα,n to the algebra. Theorem: The elements in Uloc xα,n = Mα,n+1Mα−1,nM−1

α,nM−1 α,n+1 ∈ Uloc

are mutually commuting for fixed n, and have a well-defined limit as n → ∞: xα := limn→∞ xα,n. The Q-system is integrable: Theorem The “Miura operator” µn(z) = (D − zxN,n)(D − zxN−1,n) · · · (D − zx1,n) is independent of n: µn = µn+1. The proof is a “zipper proof”, using repeated application of the Q-system, starting from the right of the expression for µn.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Integrable structure

In the cluster algebra formulation, we consider the localization of the algebra: We adjoin the inverses of the generators Mα,n to the algebra. Theorem: The elements in Uloc xα,n = Mα,n+1Mα−1,nM−1

α,nM−1 α,n+1 ∈ Uloc

are mutually commuting for fixed n, and have a well-defined limit as n → ∞: xα := limn→∞ xα,n. The Q-system is integrable: Theorem The “Miura operator” µn(z) = (D − zxN,n)(D − zxN−1,n) · · · (D − zx1,n) is independent of n: µn = µn+1. The proof is a “zipper proof”, using repeated application of the Q-system, starting from the right of the expression for µn.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Integrable structure

In the cluster algebra formulation, we consider the localization of the algebra: We adjoin the inverses of the generators Mα,n to the algebra. Theorem: The elements in Uloc xα,n = Mα,n+1Mα−1,nM−1

α,nM−1 α,n+1 ∈ Uloc

are mutually commuting for fixed n, and have a well-defined limit as n → ∞: xα := limn→∞ xα,n. The Q-system is integrable: Theorem The “Miura operator” µn(z) = (D − zxN,n)(D − zxN−1,n) · · · (D − zx1,n) is independent of n: µn = µn+1. The proof is a “zipper proof”, using repeated application of the Q-system, starting from the right of the expression for µn.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Linear recursion

Write µn(z) =

N

X

j=0

(−z)jCj,nDN−j then Cj,n = Cj are independent of n: Mutually commuting Hamiltonians. Cj = ej(x1, ..., xN). Theorem: The Miura operator acting on Mn gives a linear recursion relation with coefficients which are the conserved charges:

N

X

j=1

(−1)jCjMn−j = 0. Proof: µnM1,n = · · · (D − M1,n+1M−1

1,n )M1,n = · · · (M1,n+1 − M1,n+1) = 0.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Linear recursion

Write µn(z) =

N

X

j=0

(−z)jCj,nDN−j then Cj,n = Cj are independent of n: Mutually commuting Hamiltonians. Cj = ej(x1, ..., xN). Theorem: The Miura operator acting on Mn gives a linear recursion relation with coefficients which are the conserved charges:

N

X

j=1

(−1)jCjMn−j = 0. Proof: µnM1,n = · · · (D − M1,n+1M−1

1,n )M1,n = · · · (M1,n+1 − M1,n+1) = 0.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Translation operators

The conserved charges act as translation operators on Mn. Example: [C1, Mn] = (1 − q)Mn+1. In general use generating the function C(z) =

N

X

j=0

(−z)jCj =

N

Y

α=1

(1 − zxα) to change basis to conserved charges pk: C(z) = exp @− X

k≥1

zk k pk(x) 1 A . Then pk(x) are the power-sum symmetric functions. Theorem: [pk, Mn] = (1 − qk)Mn+k.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Translation operators

The conserved charges act as translation operators on Mn. Example: [C1, Mn] = (1 − q)Mn+1. In general use generating the function C(z) =

N

X

j=0

(−z)jCj =

N

Y

α=1

(1 − zxα) to change basis to conserved charges pk: C(z) = exp @− X

k≥1

zk k pk(x) 1 A . Then pk(x) are the power-sum symmetric functions. Theorem: [pk, Mn] = (1 − qk)Mn+k.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Translation operators

The conserved charges act as translation operators on Mn. Example: [C1, Mn] = (1 − q)Mn+1. In general use generating the function C(z) =

N

X

j=0

(−z)jCj =

N

Y

α=1

(1 − zxα) to change basis to conserved charges pk: C(z) = exp @− X

k≥1

zk k pk(x) 1 A . Then pk(x) are the power-sum symmetric functions. Theorem: [pk, Mn] = (1 − qk)Mn+k.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

How to use the Hamiltonian

To prove the exchange relation [Mn, Mn+p]q + [Mn+p−1, Mn+1]q = 0 use [C1, [Mn, Mn+ℓ]q] = (1 − q)([Mn+1, Mn+ℓ]q + [Mn, Mn+ℓ+1]q) and use induction from the commutation relation and the Q-system (ℓ = 1, ℓ = 2).

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Other generators in the algebra

The following is an automorphism of the relations (Q-system, commutation, vanishing of the quantum determinant): σ(Mα,n) = A−nMN−α,n∆−1, σ(A) = A−1, σ(∆) = ∆−1, σ(q) = q. Define f (z) = σ(e(z−1)) = X

n∈Z

(q

1 2 z−1A−1)nMN−1,n∆−1.

Then (w − zq)f (z)f (w) + (z − qw)f (w)f (z) = 0. Define ψ±(z) to be the positive (negative) powers of z in the commutation relation [e(z), f (w)] = 1 1 − q δ(z/w) ` ψ+(z) − ψ−(w) ´ .

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Other generators in the algebra

The following is an automorphism of the relations (Q-system, commutation, vanishing of the quantum determinant): σ(Mα,n) = A−nMN−α,n∆−1, σ(A) = A−1, σ(∆) = ∆−1, σ(q) = q. Define f (z) = σ(e(z−1)) = X

n∈Z

(q

1 2 z−1A−1)nMN−1,n∆−1.

Then (w − zq)f (z)f (w) + (z − qw)f (w)f (z) = 0. Define ψ±(z) to be the positive (negative) powers of z in the commutation relation [e(z), f (w)] = 1 1 − q δ(z/w) ` ψ+(z) − ψ−(w) ´ .

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Other generators in the algebra

The following is an automorphism of the relations (Q-system, commutation, vanishing of the quantum determinant): σ(Mα,n) = A−nMN−α,n∆−1, σ(A) = A−1, σ(∆) = ∆−1, σ(q) = q. Define f (z) = σ(e(z−1)) = X

n∈Z

(q

1 2 z−1A−1)nMN−1,n∆−1.

Then (w − zq)f (z)f (w) + (z − qw)f (w)f (z) = 0. Define ψ±(z) to be the positive (negative) powers of z in the commutation relation [e(z), f (w)] = 1 1 − q δ(z/w) ` ψ+(z) − ψ−(w) ´ .

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Cartan currents and conserved charges

Recall the generating function for Hamiltonians C(z) =

N

X

j=0

(−z)jCj =

N

Y

k=1

(1 − zxj) Theorem: The Cartan currents ψ±(z) are equal as rational functions of z, where ψ+(z) = (−q− 1

2 z)NAC(q 1 2 z)−1C(q− 1 2 z)−1 ∈ U[[z]]

has valuation zN, and ψ−(z) has valuation z−N.

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Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion

Conclusion

The generalized Macdonald operators give a t deformation of the quantum Q-system cluster variables. More on the connection to DAHA and elliptic Hall algebra: See talk by Di Francesco 10:30am Wednesday. The difference operators Mα,1 acting on the function 1 create q-Whittaker functions. The difference operators Mα,n generate “generalized” q-Whittaker functions: Characters of fusion products of Kirillov-Reshetikhin modules. To be understood: How to generate a representation with non-zero central charge.

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