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 sl 2 has generators e , f , h with relations. [ e , f ] = h ; [ h , e ] = 2 e ; [ h , f ] = − 2 f . The Lie algebra b sl 2 ≃ sl 2 ⊗ C [ u , u − 1 ] ⊕ C c has generators { e [ n ] , f [ n ] , h [ n ] : n ∈ Z } . Relations expressed using generating functions (currents): X X X e [ n ] z n , f [ n ] z n , h [ n ] z n . e ( z ) = f ( z ) = h ( z ) = n ∈ Z n ∈ Z n ∈ Z The key function when working with generating functions: The delta function X z − 1 1 z n = δ ( z ) = 1 − z + 1 − z − 1 . n Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion Current algebras The Lie algebra sl 2 has generators e , f , h with relations. [ e , f ] = h ; [ h , e ] = 2 e ; [ h , f ] = − 2 f . The Lie algebra b sl 2 ≃ sl 2 ⊗ C [ u , u − 1 ] ⊕ C c has generators { e [ n ] , f [ n ] , h [ n ] : n ∈ Z } . Relations expressed using generating functions (currents): X X X e [ n ] z n , f [ n ] z n , h [ n ] z n . e ( z ) = f ( z ) = h ( z ) = n ∈ Z n ∈ Z n ∈ Z The key function when working with generating functions: The delta function X z − 1 1 z n = δ ( z ) = 1 − z + 1 − z − 1 . n Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion Current algebras The Lie algebra sl 2 has generators e , f , h with relations. [ e , f ] = h ; [ h , e ] = 2 e ; [ h , f ] = − 2 f . The Lie algebra b sl 2 ≃ sl 2 ⊗ C [ u , u − 1 ] ⊕ C c has generators { e [ n ] , f [ n ] , h [ n ] : n ∈ Z } . Relations expressed using generating functions (currents): X X X e [ n ] z n , f [ n ] z n , h [ n ] z n . e ( z ) = f ( z ) = h ( z ) = n ∈ Z n ∈ Z n ∈ Z The key function when working with generating functions: The delta function X z − 1 1 z n = δ ( z ) = 1 − z + 1 − z − 1 . n Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion Current algebras The Lie algebra sl 2 has generators e , f , h with relations. [ e , f ] = h ; [ h , e ] = 2 e ; [ h , f ] = − 2 f . The Lie algebra b sl 2 ≃ sl 2 ⊗ C [ u , u − 1 ] ⊕ C c has generators { e [ n ] , f [ n ] , h [ n ] : n ∈ Z } . Relations expressed using generating functions (currents): X X X e [ n ] z n , f [ n ] z n , h [ n ] z n . e ( z ) = f ( z ) = h ( z ) = n ∈ Z n ∈ Z n ∈ Z The key function when working with generating functions: The delta function X z − 1 1 z n = δ ( z ) = 1 − z + 1 − z − 1 . n Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion Drinfeld-type relations The relations for affine sl 2 in terms of generating functions: [ h ( z ) , e ( w )] = 2 δ ( z / w ) e ( z ) , [ h ( z ) , f ( w )] = − 2 δ ( 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 ] + n δ n , − m c . Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion Drinfeld-type relations The relations for affine sl 2 in terms of generating functions: [ h ( z ) , e ( w )] = 2 δ ( z / w ) e ( z ) , [ h ( z ) , f ( w )] = − 2 δ ( 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 ] + n δ n , − m c . Kedem University of Illinois

Current algebras Difference operators Whittaker limit quantum Q-systems Conclusion Ding-Iohara deformations of U ( � sl 2 ) Fix a function g ∈ F ( z , w ), and define the universal enveloping algebra U = U g 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 ` ´ 1 Ψ + ( z ) − Ψ − ( w ) [ E ( z ) , F ( w )] = g (1 , 1) δ ( z / 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 ( � sl 2 ) Fix a function g ∈ F ( z , w ), and define the universal enveloping algebra U = U g 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 ` ´ 1 Ψ + ( z ) − Ψ − ( w ) [ E ( z ) , F ( w )] = g (1 , 1) δ ( z / 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 ` ´ 1 Ψ + ( z ) − Ψ − ( w ) [ E ( z ) , F ( w )] = g (1 , 1) δ ( z / w ) When g ( z ) = ( z − q 2 w ), the relations define the algebra U q ( b sl 2 ) at zero central charge. When g ( z ) = ( z − qw )( z − t − 1 w )( z − q − 1 tw ) the algebra is the “Ding-Iohara” algebra. If we add a “Serre relation” „ z 2 « Sym z i z 3 [ E ( z 1 ) , [ E ( z 2 ) , E ( z 3 )]] = 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 ` ´ 1 Ψ + ( z ) − Ψ − ( w ) [ E ( z ) , F ( w )] = g (1 , 1) δ ( z / w ) When g ( z ) = ( z − q 2 w ), the relations define the algebra U q ( b sl 2 ) at zero central charge. When g ( z ) = ( z − qw )( z − t − 1 w )( z − q − 1 tw ) the algebra is the “Ding-Iohara” algebra. If we add a “Serre relation” „ z 2 « Sym z i z 3 [ E ( z 1 ) , [ E ( z 2 ) , E ( z 3 )]] = 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 ` ´ 1 Ψ + ( z ) − Ψ − ( w ) [ E ( z ) , F ( w )] = g (1 , 1) δ ( z / w ) When g ( z ) = ( z − q 2 w ), the relations define the algebra U q ( b sl 2 ) at zero central charge. When g ( z ) = ( z − qw )( z − t − 1 w )( z − q − 1 tw ) the algebra is the “Ding-Iohara” algebra. If we add a “Serre relation” „ z 2 « Sym z i z 3 [ E ( z 1 ) , [ E ( z 2 ) , E ( z 3 )]] = 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 U tor ( b gl 1 ) was introduced by Miki (2000 and 2007). Also known as the deformed W 1+ ∞ -algebra. Schiffmann proved the equivalence to the elliptic Hall algebra. Schiffmann-Vasserot gave an isomorphism: Elliptic Hall ≃ spherical DAHA of type A N when N → ∞ . The limit t → ∞ of the quantum toroidal algebra is a quantum affine algebra (after renormalization of generators). U √ q ( b sl 2 ) or half of it: U √ q ( n + [ u , u − 1 ]). [Hall algebra, Kapranov]. Kedem University of Illinois