Logarithmic Frobenius structures and Coxeter discriminants

Feigin, Misha; Веселов, А. П.

doi:10.1016/j.aim.2006.08.010

Cited by 35 publications

(107 citation statements)

References 17 publications

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“…The restrictions of Coxeter root systems appear also, in particular, in the context of ∨-systems [36]. We note that the number of vectors in the ∨-system (H 4 , A 1 ) is stated inaccurately in [36] as some of the vectors listed there are actually proportional.…”

Section: Propositionmentioning

confidence: 99%

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Generalized Calogero–Moser systems from rational Cherednik algebras

Feigin

2011

Sel. Math. New Ser.

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We consider ideals of polynomials vanishing on the W -orbits of the intersections of mirrors of a finite reflection group W . We determine all such ideals that are invariant under the action of the corresponding rational Cherednik algebra hence form submodules in the polynomial module. We show that a quantum integrable system can be defined for every such ideal for a real reflection group W . This leads to known and new integrable systems of Calogero-Moser type which we explicitly specify. In the case of classical Coxeter groups, we also obtain generalized CalogeroMoser systems with added quadratic potential.

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Section: Propositionmentioning

confidence: 99%

“…We note that the number of vectors in the ∨-system (H 4 , A 1 ) is stated inaccurately in [36] as some of the vectors listed there are actually proportional. We also refer to [37] where, in particular, bases of the restricted Weyl root systems are discussed.…”

Section: Propositionmentioning

confidence: 99%

Generalized Calogero–Moser systems from rational Cherednik algebras

Feigin

2011

Sel. Math. New Ser.

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“…Repeating the argument for the terms that appear in Eqs. (13) and (14) yields no further conditions. 2…”

Section: It Is Easy To Show That ξ a = 0 If And Only If H β = H σ α βmentioning

confidence: 91%

Weyl groups and elliptic solutions of the WDVV equations

Strachan

2010

Advances in Mathematics

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A functional ansatz is developed which gives certain elliptic solutions of the Witten-Dijkgraaf-VerlindeVerlinde (or WDVV) equations. This ansatz is based on the elliptic trilogarithm function introduced by Beilinson and Levin. For this to be a solution results in a number of purely algebraic conditions on the set of vectors that appear in the ansatz, this providing an elliptic version of the idea, introduced by Veselov, of a ∨-system.Rational and trigonometric limits are studied together with examples of elliptic ∨-systems based on various Weyl groups. Jacobi group orbit spaces are studied: these carry the structure of a Frobenius manifold. The corresponding 'almost dual' structure is shown, in the A N and B N cases and conjecturally for an arbitrary Weyl group, to correspond to the elliptic solutions of the WDVV equations.Transformation properties, under the Jacobi group, of the elliptic trilogarithm are derived together with various functional identities which generalize the classical Frobenius-Stickelberger relations.

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“…It would be interesting to obtain almost dual prepotentials for the Frobenius manifolds of the affine Weyl groups as well as for their discriminants (cf. rational case [7,12]). Comparison with a recent work on the elliptic solutions [16] might also be interesting.…”

Section: Discussionmentioning

confidence: 99%

Trigonometric Solutions of WDVV Equations and Generalized Calogero-Moser-Sutherland Systems

Feigin¹

2009

SIGMA

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Abstract. We consider trigonometric solutions of WDVV equations and derive geometric conditions when a collection of vectors with multiplicities determines such a solution. We incorporate these conditions into the notion of trigonometric Veselov system (∨-system) and we determine all trigonometric ∨-systems with up to five vectors. We show that generalized Calogero-Moser-Sutherland operator admits a factorized eigenfunction if and only if it corresponds to the trigonometric ∨-system; this inverts a one-way implication observed by Veselov for the rational solutions.

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Logarithmic Frobenius structures and Coxeter discriminants

Cited by 35 publications

References 17 publications

Generalized Calogero–Moser systems from rational Cherednik algebras

Generalized Calogero–Moser systems from rational Cherednik algebras

Weyl groups and elliptic solutions of the WDVV equations

Trigonometric Solutions of WDVV Equations and Generalized Calogero-Moser-Sutherland Systems

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