On the complete integrability of the periodic quantum Toda lattice

Mare, Augustin-Liviu

doi:10.1515/forum-2016-0117

Cited by 1 publication

(5 citation statements)

References 14 publications

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“…This result was proved by Goodman and Wallach for types

A_{n} - D_{n}

and

E_{6}

, and by Mare in the types

F_{4}

and

G_{2}

. Etingof proved earlier the integrability of the (non‐dual) quantum periodic Toda lattice, for all Lie types, and this implies the above result for the simply laced Lie algebras.…”

Section: The Periodic Toda Lattice and Relations In Prefixqh Aff ∗Falmentioning

confidence: 66%

“…Such operators were constructed by Etingof in the non‐dual case and all Lie types, and in the dual case by Goodman and Wallach for

G

of Lie types

A_{n} - D_{n}

and

E_{6}

. In , Mare used Dynkin automorphisms and the results from to construct such operators in the remaining Lie types

F_{4}

and

G_{2}

. This is consistent to the philosophy of B. Kim that the relations in quantum cohomology of

G / B

are given by integrals of motion for the Toda lattice of the Langlands dual root system.…”

Section: The Periodic Toda Lattice and Relations In Prefixqh Aff ∗Falmentioning

confidence: 77%

“…The complete integrability of this system has been established by Goodman and Wallach for

G

of Lie types

A_{n}, B_{n}, C_{n}, D_{n}

and

E_{6}

, and by Etingof in the simply laced Lie types (in these cases, there is no difference between the dual and the non‐dual versions of the periodic quantum Toda lattice). For the remaining types

F_{4}

and

G_{2}

the integrability has been proved by Mare .…”

Section: Introductionmentioning

confidence: 89%

“…Theorem Fix

i \in {1, \dots, n}

. There exists a unique

{normalΩ}_{i} \in U {(b)}_{e v}

such that

[{normalΩ}_{i}, Ω] = 0;

Ω_{i}

has degree equal to

deg f_{i}

; the symbol

μ false(Ω_{i} false) = f_{i}

is the chosen generator of

Sym {({frakturh}^{*})}^{W}

. Further, for

{normalΩ}_{1} : = Ω

and for any

1 ⩽ i, j ⩽ n

[{normalΩ}_{i}, {normalΩ}_{j}] = 0

.…”

Section: The Periodic Toda Lattice and Relations In Prefixqh Aff ∗Falmentioning

confidence: 99%

“…Etingof proved earlier the integrability of the (non‐dual) quantum periodic Toda lattice, for all Lie types, and this implies the above result for the simply laced Lie algebras. We refer to for details.…”

Section: The Periodic Toda Lattice and Relations In Prefixqh Aff ∗Falmentioning

confidence: 99%

See 4 more Smart Citations

An affine quantum cohomology ring for flag manifolds and the periodic Toda lattice

Mare

Mihalcea

2017

Proc. London Math. Soc.

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Consider the generalized flag manifold G/B and the corresponding affine flag manifold F G. In this paper we use curve neighborhoods for Schubert varieties in F G to construct certain affine Gromov-Witten invariants of F G, and to obtain a family of 'affine quantum Chevalley' operators Λ0, . . . , Λn indexed by the simple roots in the affine root system of G. These operators act on the cohomology ring H * (F G) with coefficients in Z[q0, . . . , qn]. By analyzing commutativity and invariance properties of these operators we deduce the existence of two quantum cohomology rings, which satisfy properties conjectured earlier by Guest and Otofuji for G = SLn(C). The first quantum ring is a deformation of the subalgebra of H * (F G) generated by divisors. The second ring, denoted QH * aff (G/B), deforms the ordinary quantum cohomology ring QH * (G/B) by adding an affine quantum parameter q0. We prove that QH * aff (G/B) is a Frobenius algebra, and that the new quantum product determines a flat Dubrovin connection. Further, we develop an analogue of Givental and Kim formalism for this ring and we deduce a presentation of QH * aff (G/B) by generators and relations. The ideal of relations is generated by the integrals of motion for the periodic Toda lattice associated to the dual of the extended Dynkin diagram of G.Contents Schubert classes ε 0 , . . . , ε n . The main result of this paper, proved in § 7, is the following.Theorem 1.1. The affine quantum Chevalley operators Λ i satisfy the following properties.(a) The operators Λ i commute up to the imaginary coroot, that is, for any w ∈ W aff and any 0 i, j n we haveThe modified operators Λ i − m i Λ 0 commute (without any additional constraint), that is, for any w ∈ W aff and any 1 i, j n we have

show abstract

“…This result was proved by Goodman and Wallach for types

A_{n} - D_{n}

and

E_{6}

, and by Mare in the types

F_{4}

and

G_{2}

. Etingof proved earlier the integrability of the (non‐dual) quantum periodic Toda lattice, for all Lie types, and this implies the above result for the simply laced Lie algebras.…”

Section: The Periodic Toda Lattice and Relations In Prefixqh Aff ∗Falmentioning

confidence: 66%

“…Such operators were constructed by Etingof in the non‐dual case and all Lie types, and in the dual case by Goodman and Wallach for

G

of Lie types

A_{n} - D_{n}

and

E_{6}

. In , Mare used Dynkin automorphisms and the results from to construct such operators in the remaining Lie types

F_{4}

and

G_{2}

. This is consistent to the philosophy of B. Kim that the relations in quantum cohomology of

G / B

are given by integrals of motion for the Toda lattice of the Langlands dual root system.…”

Section: The Periodic Toda Lattice and Relations In Prefixqh Aff ∗Falmentioning

confidence: 77%

“…The complete integrability of this system has been established by Goodman and Wallach for

G

of Lie types

A_{n}, B_{n}, C_{n}, D_{n}

and

E_{6}

, and by Etingof in the simply laced Lie types (in these cases, there is no difference between the dual and the non‐dual versions of the periodic quantum Toda lattice). For the remaining types

F_{4}

and

G_{2}

the integrability has been proved by Mare .…”

Section: Introductionmentioning

confidence: 89%

“…Theorem Fix

i \in {1, \dots, n}

. There exists a unique

{normalΩ}_{i} \in U {(b)}_{e v}

such that

[{normalΩ}_{i}, Ω] = 0;

Ω_{i}

has degree equal to

deg f_{i}

; the symbol

μ false(Ω_{i} false) = f_{i}

is the chosen generator of

Sym {({frakturh}^{*})}^{W}

. Further, for

{normalΩ}_{1} : = Ω

and for any

1 ⩽ i, j ⩽ n

[{normalΩ}_{i}, {normalΩ}_{j}] = 0

.…”

Section: The Periodic Toda Lattice and Relations In Prefixqh Aff ∗Falmentioning

confidence: 99%

Section: The Periodic Toda Lattice and Relations In Prefixqh Aff ∗Falmentioning

confidence: 99%

See 3 more Smart Citations

An affine quantum cohomology ring for flag manifolds and the periodic Toda lattice

Mare

Mihalcea

2017

Proc. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

On the complete integrability of the periodic quantum Toda lattice

Cited by 1 publication

References 14 publications

An affine quantum cohomology ring for flag manifolds and the periodic Toda lattice

An affine quantum cohomology ring for flag manifolds and the periodic Toda lattice

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