Comultiplication for shifted Yangians and quantum open Toda lattice

Finkelberg, Michael; Kamnitzer, Joel; Pham, Khoa; Rybnikov, Leonid; Weekes, Alex

doi:10.1016/j.aim.2017.06.018

Cited by 30 publications

(49 citation statements)

References 22 publications

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“…Fix a pair of coweights µ 1 , µ 2 such that µ 1 + µ 2 = µ. Following [FKPRW,§5.4], consider the filtration F • µ 1 ,µ 2 Y µ of Y µ by defining degrees of the PBW generators as follows:…”

Section: )mentioning

confidence: 99%

“…Another definition of the shifted Yangian (for an arbitrary, not necessarily dominant, shift) as a Rees algebra was given in [FKPRW,§5.4] and [BFNb, Appendix B(i)], precisely to avoid the issues mentioned above. This raises Question: Are these two definitions of the shifted Yangians equivalent for dominant shifts?…”

Section: By Alexander Tsymbaliuk and Alex Weekesmentioning

confidence: 99%

“…Choose any total ordering on the set of PBW generators as in (2.54) (but generalized from type A to any g, cf. [FKPRW,(3.4)]). The following is analogous to [FKPRW,Corollary 3.15] (cf.…”

Section: Hence An Embedding Of the Rees Algebrasmentioning

confidence: 99%

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Shifted Quantum Affine Algebras: Integral Forms in Type A

Finkelberg

Tsymbaliuk

2019

Arnold Math J.

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We define an integral form of shifted quantum affine algebras of type A and construct Poincaré-Birkhoff-Witt-Drinfeld bases for them. When the shift is trivial, our integral form coincides with the RTT integral form. We prove that these integral forms are closed with respect to the coproduct and shift homomorphisms. We prove that the homomorphism from our integral form to the corresponding quantized K-theoretic Coulomb branch of a quiver gauge theory is always surjective. In one particular case we identify this Coulomb branch with the extended quantum universal enveloping algebra of type A. Finally, we obtain the rational (homological) analogues of the above results (proved earlier in [KMWY, KTWWY] via different techniques).

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Section: )mentioning

confidence: 99%

Section: By Alexander Tsymbaliuk and Alex Weekesmentioning

confidence: 99%

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Shifted Quantum Affine Algebras: Integral Forms in Type A

Finkelberg

Tsymbaliuk

2019

Arnold Math J.

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“…Our interest in

Y_{μ}^{λ}

comes from deformation quantization. When

frakturg

is finite type, there is a filtration on

Y_{μ}

, which is defined explicitly in terms of a PBW basis, see [, Appendix B(i)] or [, Section 5.4]. Consider the corresponding quotient filtration on

Y_{μ}^{λ}

.…”

Section: Truncated Shifted Yangians and Klr Yangian Algebrasmentioning

confidence: 99%

On category O for affine Grassmannian slices and categorified tensor products

Kamnitzer

Tingley

Webster

et al. 2019

Proc. London Math. Soc.

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Truncated shifted Yangians are a family of algebras which naturally quantize slices in the affine Grassmannian. These algebras depend on a choice of two weights λ and μ for a Lie algebra frakturg, which we will assume is simply laced. In this paper, we relate the category scriptO over truncated shifted Yangians to categorified tensor products: For a generic integral choice of parameters, category scriptO is equivalent to a weight space in the categorification of a tensor product of fundamental representations defined by the third author using KLRW algebras. We also give a precise description of category scriptO for arbitrary parameters using a new algebra which we call the parity KLRW algebra. In particular, we confirm the conjecture of the authors that the highest weights of category scriptO are in canonical bijection with a product monomial crystal depending on the choice of parameters. This work also has interesting applications to classical representation theory. In particular, it allows us to give a classification of simple Gelfand–Tsetlin modules of U(frakturgln) and its associated W‐algebras.

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“…Proposition 7.2. [FKPRW,5.9] The multiplication morphism W λ 1 µ 1 × W λ 2 µ 2 → W λ 1 +λ 2 µ 1 +µ 2 is given by (g 1 , g 2 ) → π(g 1 g 2 ), g 1 ∈ W…”

mentioning

confidence: 99%

Integrable Crystals and Restriction to Levi Subgroups Via Generalized Slices in the Affine Grassmannian

Krylov

2018

Funct Anal Its Appl

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Let G be a connected reductive algebraic group over C. Let Λ + G be the monoid of dominant weights of G. We construct the integrable crystals B G (λ), λ ∈ Λ + G , using the geometry of generalized transversal slices in the affine Grassmannian of the Langlands dual group. We construct the tensor product maps p λ 1 ,λ 2 :in terms of multiplication of generalized transversal slices. Let L ⊂ G be a Levi subgroup of G. We describe the restriction to Levi Res G L : Rep(G) → Rep(L) in terms of the hyperbolic localization functors for the generalized transversal slices.

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Comultiplication for shifted Yangians and quantum open Toda lattice

Cited by 30 publications

References 22 publications

Shifted Quantum Affine Algebras: Integral Forms in Type A

Shifted Quantum Affine Algebras: Integral Forms in Type A

On category O for affine Grassmannian slices and categorified tensor products

Integrable Crystals and Restriction to Levi Subgroups Via Generalized Slices in the Affine Grassmannian

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