Duality for integrable systems associated to quantum toroidal - PowerPoint PPT Presentation

Duality for integrable systems associated to quantum toroidal algebras Evgeny Mukhin Indiana University Purdue University Indianapolis XIX International Congress on Mathematical Physics Montreal, July 2018 Evgeny Mukhin (IUPUI) Duality for quantum toroidal algebras Montreal, July 2018 1 / 16

The algebras of quantum Hamiltonians The transfer matrices Let U q be your favorite quantum group. Let R ∈ U q ˜ ⊗ U q be the R -matrix satisfying the Yang-Baxter equation R 12 R 13 R 23 = R 23 R 13 R 12 . Let Q ∈ U q be the twist operator: R ( Q ⊗ Q ) = ( Q ⊗ Q ) R . Let V be an admissible U q -module. Then the trace T V = (Tr V ⊗ 1) (( Q ⊗ 1) R ) ∈ ˜ U q is called the transfer matrix. Lemma. For any admissible modules V 1 , V 2 , the transfer matrices commute: T V 1 T V 2 = T V 2 T V 1 . Evgeny Mukhin (IUPUI) Duality for quantum toroidal algebras Montreal, July 2018 2 / 16

The algebras of quantum Hamiltonians The XXZ type models Recall : T V = (Tr V ⊗ 1) (( Q ⊗ 1) R ) . Thus, the R matrix gives an embedding of the Grothendick ring of admissible representations to the quantum group: T : K 0 ( Rep U q ) → ˜ U q , V �→ T V . The image B q = Im( T ) is the commutative algebra of quantum Hamiltonians. The algebra B q acts on an appropriate class of representations of U q . Problem. (XXZ type models) Understand the spectrum of B q . Evgeny Mukhin (IUPUI) Duality for quantum toroidal algebras Montreal, July 2018 3 / 16

The algebras of quantum Hamiltonians The Gaudin type models The limit q → 1 gives an algebra of quantum Hamiltonians in the corresponding universal enveloping algebra: q → 1 B q ∈ ˜ B = lim U. The limit is not easy. There are alternative constructions (of the same algebra) for affine Lie algebras: from the center on the critical level [FFR]; from Segal-Sugawara vectors in the vacuum modules [M]; shift of argument method [R]. The algebra B acts on an appropriate class of representations of U . Problem. (Gaudin type models) Understand the spectrum of B . Evgeny Mukhin (IUPUI) Duality for quantum toroidal algebras Montreal, July 2018 4 / 16

The algebras of quantum Hamiltonians The Gaudin type models The limit q → 1 gives an algebra of quantum Hamiltonians in the corresponding universal enveloping algebra: q → 1 B q ∈ ˜ B = lim U. The limit is not easy. There are alternative constructions (of the same algebra) for affine Lie algebras: B.Feigin, E.Frenkel, and from the center on the critical level [FFR]; N. Reshetikhin, (94) from Segal-Sugawara vectors in the vacuum modules [M]; shift of argument method [R]. The algebra B acts on an appropriate class of representations of U . Problem. (Gaudin type models) Understand the spectrum of B . Evgeny Mukhin (IUPUI) Duality for quantum toroidal algebras Montreal, July 2018 4 / 16

The algebras of quantum Hamiltonians The Gaudin type models The limit q → 1 gives an algebra of quantum Hamiltonians in the corresponding universal enveloping algebra: q → 1 B q ∈ ˜ B = lim U. The limit is not easy. There are alternative constructions (of the same algebra) for affine Lie algebras: from the center on the critical level [FFR]; from Segal-Sugawara vectors in the vacuum modules [M]; shift of argument method [R]. The algebra B acts on an appropriate class of representations of U . Problem. (Gaudin type models) Understand the spectrum of B . Evgeny Mukhin (IUPUI) Duality for quantum toroidal algebras Montreal, July 2018 4 / 16

The algebras of quantum Hamiltonians The Gaudin type models The limit q → 1 gives an algebra of quantum Hamiltonians in the corresponding universal enveloping algebra: q → 1 B q ∈ ˜ B = lim U. The limit is not easy. There are alternative constructions (of the same algebra) for affine Lie algebras: from the center on the critical level [FFR]; A. Molev, (11) from Segal-Sugawara vectors in the vacuum modules [M]; shift of argument method [R]. The algebra B acts on an appropriate class of representations of U . Problem. (Gaudin type models) Understand the spectrum of B . Evgeny Mukhin (IUPUI) Duality for quantum toroidal algebras Montreal, July 2018 4 / 16

The algebras of quantum Hamiltonians The Gaudin type models The limit q → 1 gives an algebra of quantum Hamiltonians in the corresponding universal enveloping algebra: q → 1 B q ∈ ˜ B = lim U. The limit is not easy. There are alternative constructions (of the same algebra) for affine Lie algebras: from the center on the critical level [FFR]; from Segal-Sugawara vectors in the vacuum modules [M]; shift of argument method [R]. The algebra B acts on an appropriate class of representations of U . Problem. (Gaudin type models) Understand the spectrum of B . Evgeny Mukhin (IUPUI) Duality for quantum toroidal algebras Montreal, July 2018 4 / 16

The algebras of quantum Hamiltonians The Gaudin type models The limit q → 1 gives an algebra of quantum Hamiltonians in the corresponding universal enveloping algebra: q → 1 B q ∈ ˜ B = lim U. The limit is not easy. There are alternative constructions (of the same algebra) for affine Lie algebras: from the center on the critical level [FFR]; from Segal-Sugawara vectors in the vacuum modules [M]; L. Rybnikov, (06) shift of argument method [R]. The algebra B acts on an appropriate class of representations of U . Problem. (Gaudin type models) Understand the spectrum of B . Evgeny Mukhin (IUPUI) Duality for quantum toroidal algebras Montreal, July 2018 4 / 16

The algebras of quantum Hamiltonians The Gaudin type models The limit q → 1 gives an algebra of quantum Hamiltonians in the corresponding universal enveloping algebra: q → 1 B q ∈ ˜ B = lim U. The limit is not easy. There are alternative constructions (of the same algebra) for affine Lie algebras: from the center on the critical level [FFR]; from Segal-Sugawara vectors in the vacuum modules [M]; shift of argument method [R]. The algebra B acts on an appropriate class of representations of U . Problem. (Gaudin type models) Understand the spectrum of B . Evgeny Mukhin (IUPUI) Duality for quantum toroidal algebras Montreal, July 2018 4 / 16

The algebras of quantum Hamiltonians An example Let U = gl n [ t ] = gl n ⊗ C [ t ] . ∞ � ( e ij ⊗ t s ) x − s − 1 ∈ U [[ x − 1 ]] . We use formal series e ij ( x ) = n � s =0 Let ¯ Q = u i e ii . i =1 Consider the matrix   ∂ x − u 1 − e 11 ( x ) − e 21 ( x ) . . . − e n 1 ( x )   − e 12 ( x ) ∂ x − u 2 − e 22 ( x ) . . . − e n 2 ( x ) E u   n =  .  . . . . . . . . . . . . − e 1 n ( x ) − e 2 n ( x ) . . . ∂ x − u n − e nn ( x ) Expand the row determinant: rdet E u n = ∂ n x + B 1 ( x ) ∂ n − 1 + B 2 ( x ) ∂ n − 2 + · · · + B n ( x ) . x x Theorem. ([T]) Coefficients of B i ( x ) commute and generate the algebra B u n of quantum Hamiltonians in gl n [ t ] . Evgeny Mukhin (IUPUI) Duality for quantum toroidal algebras Montreal, July 2018 5 / 16

The algebras of quantum Hamiltonians An example Let U = gl n [ t ] = gl n ⊗ C [ t ] . ∞ � ( e ij ⊗ t s ) x − s − 1 ∈ U [[ x − 1 ]] . We use formal series e ij ( x ) = n � s =0 Let ¯ Q = u i e ii . i =1 Consider the matrix   ∂ x − u 1 − e 11 ( x ) − e 21 ( x ) . . . − e n 1 ( x )   − e 12 ( x ) ∂ x − u 2 − e 22 ( x ) . . . − e n 2 ( x ) E u   n =  .  . . . . . . . . . . . . − e 1 n ( x ) − e 2 n ( x ) . . . ∂ x − u n − e nn ( x ) Expand the row determinant: rdet E u n = ∂ n x + B 1 ( x ) ∂ n − 1 + B 2 ( x ) ∂ n − 2 + · · · + B n ( x ) . x x Theorem. ([T]) Coefficients of B i ( x ) commute and generate the algebra B u n of D. Talalaev, (04) quantum Hamiltonians in gl n [ t ] . Evgeny Mukhin (IUPUI) Duality for quantum toroidal algebras Montreal, July 2018 5 / 16

The algebras of quantum Hamiltonians An example Let U = gl n [ t ] = gl n ⊗ C [ t ] . ∞ � ( e ij ⊗ t s ) x − s − 1 ∈ U [[ x − 1 ]] . We use formal series e ij ( x ) = n � s =0 Let ¯ Q = u i e ii . i =1 Consider the matrix   ∂ x − u 1 − e 11 ( x ) − e 21 ( x ) . . . − e n 1 ( x )   − e 12 ( x ) ∂ x − u 2 − e 22 ( x ) . . . − e n 2 ( x ) E u   n =  .  . . . . . . . . . . . . − e 1 n ( x ) − e 2 n ( x ) . . . ∂ x − u n − e nn ( x ) Expand the row determinant: rdet E u n = ∂ n x + B 1 ( x ) ∂ n − 1 + B 2 ( x ) ∂ n − 2 + · · · + B n ( x ) . x x Theorem. ([T]) Coefficients of B i ( x ) commute and generate the algebra B u n of quantum Hamiltonians in gl n [ t ] . Evgeny Mukhin (IUPUI) Duality for quantum toroidal algebras Montreal, July 2018 5 / 16

Dualities The classical gl m − gl n duality Consider the vector space V = C [ x ij ] j =1 ,...,n i =1 ,...,m . Evgeny Mukhin (IUPUI) Duality for quantum toroidal algebras Montreal, July 2018 6 / 16

Dualities The classical gl m − gl n duality Consider the vector space V = C [ x ij ] j =1 ,...,n i =1 ,...,m .   x 11 x 12 . . . x 1 n   x 21 x 22 . . . x 2 n     . . . . . . . . . . . . x m 1 x m 2 . . . x mn Evgeny Mukhin (IUPUI) Duality for quantum toroidal algebras Montreal, July 2018 6 / 16

Dualities The classical gl m − gl n duality Consider the vector space V = C [ x ij ] j =1 ,...,n i =1 ,...,m .   x 11 x 12 . . . x 1 n gl m gl n   x 21 x 22 . . . x 2 n     . . . . . . . . . . . . x m 1 x m 2 . . . x mn Evgeny Mukhin (IUPUI) Duality for quantum toroidal algebras Montreal, July 2018 6 / 16