Multiplicative Slices, Relativistic Toda and Shifted Quantum Affine Algebras

Finkelberg, Michael; Tsymbaliuk, Alexander

doi:10.1007/978-3-030-23531-4_6

Cited by 34 publications

(49 citation statements)

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“…In Section 3(ii), we recall the RTT integral form U rtt v (Lgl n ) following [FRT, DF]. The latter is a C[v, v −1 ]-algebra, which can be thought of as a quantization of the algebra of functions on the thick slice † W 0 of [FT,4(viii)] (see (3.12) and Remark 3.61) as v → 1.…”

Section: Contentsmentioning

confidence: 99%

“…In Section 4(ii), we recall the notion of the (extended) quantized K-theoretic Coulomb branch A v (which is a C[v, v −1 ]-algebra) and the fact that Φ λ µ gives rise to a homomor- In Section 4(iii), we prove a reduced version of the integral counterpart of [FT,Conjecture 8.14], see Conjecture 4.40, which identifies Ker(Φ λ µ ) with an explicit truncation ideal I λ µ in the particular case µ = 0, λ = nω n−1 (which corresponds to the dimension vector (1, 2, . .…”

Section: Contentsmentioning

confidence: 99%

“…In Section 4(iv), we prove that the C(v)-algebra homomorphisms ∆ µ 1 ,µ 2 : [FT,Theorem 10.26] generalizing the Drinfeld-Jimbo coproduct on U v (Lsl n ) give rise to C[v, v −1 ]-algebra homomorphisms ∆ µ 1 ,µ 2 :…”

Section: Contentsmentioning

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: Contentsmentioning

confidence: 99%

Section: Contentsmentioning

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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“…In [46], the Lax matrices for the Toda system were discussed in the context of 'shifted Yangians' or 'shifted quantum affine algebras'. Apparently, some of these Lax matrices have similar structures as L-operators for Q-operators.…”

Section: Discussionmentioning

confidence: 99%

A note on q-oscillator realizations of U(gl(M|N)) for Baxter Q-operators

Tsuboi

2019

Nuclear Physics B

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A note on q-oscillator realizations of U q (gl(M |N )) forBaxter Q-operators Zengo TsuboiLaboratory of physics of living matter, AbstractWe consider asymptotic limits of q-oscillator (or Heisenberg) realizations of Verma modules over the quantum superalgebra U q (gl(M |N )), and obtain qoscillator realizations of the contracted algebras proposed in [1]. Instead of factoring out the invariant subspaces, we make reduction on generators of the q-oscillator algebra, which gives a shortcut to the problem. Based on this result, we obtain explicit q-oscillator representations of a Borel subalgebra of the quantum affine superalgebra U q (ĝl(M |N )) for Baxter Q-operators.1 As for the rational (q = 1) case, see [14,15,16]. There is another approach to Q-operators [17,18,19].1 over a Borel subalgebra of U q (ŝl(3)). Moreover, Hernandez and Jimbo showed [20] that the same type of q-oscillator representations can be systematically constructed by taking asymptotic limits of Kirillov-Reshetikhin modules over one of the Borel subalgebras of any non-twisted quantum affine algebra. In addition, this approach was further developed [21,22] for U q (ŝl(M|N)) case. Hernandez and Jimbo's approach is representation theoretically sophisticated, but rather abstract, and thus it is still meaningful to seek another method to obtain explicit q-oscillator realizations, which will be useful for applications to concrete problems. In this paper, we make a proposal on this for U q (ĝl(M|N)) case, where we develop, in part, the scheme proposed in our previous paper [1]. In our classification [23] of the Q-operators, there are 2 M +N kinds of Q-operators for U q (ĝl(M|N)), each of which is labeled by a subset I of {1, 2, . . . , M + N}. In the paper [1], we mainly considered Card(I) = 0, 1, M + N − 1, M + N cases 2 . In this paper, we propose q-oscillator realizations for 2 ≤ Card(I) ≤ M + N − 2 case.In general, the Kirillov-Reshetikhin modules are considered to be derived from Verma modules based on a procedure, called the BGG-resolution. This implies that one has to factor out unnecessary invariant subspaces to get the final results if one starts from Verma modules [4,13]. In this paper, we also start from Verma modules, but realize them in terms of the q-oscillator algebra based on the Heisenberg realization (q-difference realization) of U q (gl(M|N)) [24, 25] on the flag manifold (for N = 0 case, [26]) from the very beginning, and then consider reduction on generators of the q-oscillator algebra, from which we obtain various q-oscillator realizations of U q (gl(M|N)) that interpolate the full Verma module and the simplest q-oscillator realization, namely, the q-Holstein-Primakoff type realization (cf. [27]). By taking limits of them, we obtain q-oscillator realizations of contracted algebras 3 U q (gl(M|N; I)) for U q (gl(M|N)) [1], and those of the q-super-Yangian Y q (gl(M|N)) via an evaluation map. A merit to consider reduction on the q-oscillator algebra lies in the fact that we do not have to factor out invariant subspaces, and thereby ar...

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“…The quantum toroidal algebra in the left-hand side of this relation is known ( [17]) to act on the K-theory of the following moduli space:…”

Section: Introductionmentioning

confidence: 99%

Moduli Spaces of Sheaves on Surfaces: Hecke Correspondences and Representation Theory

Neguţ

2019

Lecture Notes in Mathematics

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We construct explicit elements W k ij in (a completion of) the shifted quantum toroidal algebra of type A, and show that these elements act by 0 on the K-theory of moduli spaces of parabolic sheaves. We expect that the quotient of the shifted quantum toroidal algebra by the ideal generated by the elements W k ij will be related to q-deformed W -algebras of type A for arbitrary nilpotent, which would imply a q-deformed version of the AGT correspondence between gauge theory with surface operators and conformal field theory.Proposition 4.5. Under the action of Theorem 4.3, the elements (2.21) act on K r as the operators (4.7) with the same name, hence the abuse of notation.

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Multiplicative Slices, Relativistic Toda and Shifted Quantum Affine Algebras

Cited by 34 publications

References 47 publications

Shifted Quantum Affine Algebras: Integral Forms in Type A

Shifted Quantum Affine Algebras: Integral Forms in Type A

A note on q-oscillator realizations of U(gl(M|N)) for Baxter Q-operators

Moduli Spaces of Sheaves on Surfaces: Hecke Correspondences and Representation Theory

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