Isomorphism Between the R-Matrix and Drinfeld Presentations of Yangian in Types B, C and D

Jing, Naihuan; Liu, Ming; Molev, Alexander

doi:10.1007/s00220-018-3185-x

Cited by 43 publications

(117 citation statements)

References 19 publications

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“…The observed simple structure of the spinorial R so(4) matrix is helpful for the diagonalization of the trace of the spinorial monodromy matrix 7) or in components…”

Section: Aba For the So(4) Casementioning

confidence: 99%

“…We obtain two blocks of the sℓ(4) R matrix form. (1,5), and (γ 1 , γ 2 ) = (1, 1), (1,4), (1,6), (1,7) is solved by…”

Section: Aba For the So(6) Casementioning

confidence: 99%

“…T 5 1 (v) = T 5 4 (v) = T 5 6 (v) = T 5 7 (v) = 0. Choosing (α 1 , α 2 ) = (1, 1), (1,4), (1,6), (1,7), and (γ 1 , γ 2 ) = (1, 2), (1,3), (1,5), one deduces…”

Section: Aba For the So(6) Casementioning

confidence: 99%

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Spinorial R operator and Algebraic Bethe Ansatz

Karakhanyan

Kirschner

2020

Nuclear Physics B

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We propose a new approach to the spinor-spinor R-matrix with orthogonal and symplectic symmetry. Based on this approach and the fusion method we relate the spinorvector and vector-vector monodromy matrices for quantum spin chains. We consider the explicit spinor R matrices of low rank orthogonal algebras and the corresponding RT T algebras. Coincidences with fundamental R matrices allow to relate the Algebraic Bethe Ansatz for spinor and vector monodromy matrices. 1

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“…The observed simple structure of the spinorial R so(4) matrix is helpful for the diagonalization of the trace of the spinorial monodromy matrix 7) or in components…”

Section: Aba For the So(4) Casementioning

confidence: 99%

“…We obtain two blocks of the sℓ(4) R matrix form. (1,5), and (γ 1 , γ 2 ) = (1, 1), (1,4), (1,6), (1,7) is solved by…”

Section: Aba For the So(6) Casementioning

confidence: 99%

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Spinorial R operator and Algebraic Bethe Ansatz

Karakhanyan

Kirschner

2020

Nuclear Physics B

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“…As with the corresponding isomorphisms between the R-matrix and Drinfeld presentations of the Yangians (see respective details in [4], [16] and [19]), their counterparts in the quantum affine algebra case allow one to connect two sides of the representation theory in an explicit way: the parameterization of finite-dimensional irreducible representations via their Drinfeld polynomials can be easily translated from one presentation to another; see [5,Chapter 12], [15] and [24].…”

Section: Introductionmentioning

confidence: 99%

Isomorphism between the R-matrix and Drinfeld presentations of quantum affine algebra: Type C

Jing

Liu

Molev

2020

Journal of Mathematical Physics

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Following the approach of Ding and I. Frenkel (1993) for type A, we showed in our previous work that the Gauss decomposition of the generator matrix in the R-matrix presentation of the quantum affine algebra yields the Drinfeld generators in all classical types. Complete details for type C were given therein, while the present paper deals with types B and D. The arguments for all classical types are quite similar so we mostly concentrate on necessary additional details specific to the underlying orthogonal Lie algebras.

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“…For elliptic quantum groups associated with other simple Lie algebras, one possible first step would be to derive the RLL presentation; see [33,39] for Yangians.…”

Section: Introductionmentioning

confidence: 99%

Elliptic quantum groups and Baxter relations

Zhang

2018

Alg. Number Th.

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We introduce a category O of modules over the elliptic quantum group of sl N with well-behaved q-character theory. We construct asymptotic modules as analytic continuation of a family of finite-dimensional modules, the Kirillov-Reshetikhin modules. In the Grothendieck ring of this category we prove two types of identities: generalized Baxter relations in the spirit of Frenkel-Hernandez between finite-dimensional modules and asymptotic modules; three-term Baxter TQ relations of infinite-dimensional modules.1 2 HUAFENG ZHANG Such R-matrices R(z; λ) appeared previously in face-type integrable models [20,38]; for instance, the R-matrix of the Andrews-Baxter-Forrester model comes from two-dimensional irreducible modules of E τ, (sl 2 ), as does the 6-vertex model from the affine quantum group U (Lsl 2 ). The definition of E τ, (sl N ) in [17], by RLL exchange relations, is in the spirit of Faddeev-Reshetikhin-Takhatajan, originated from Quantum Inverse Scattering Method. We mention that elliptic R-matrices describe the monodromy of the quantized Knizhnik-Zamolodchikov equation associated with representations of affine quantum groups, e.g. [28,29,43,50].Recently Aganagic-Okounkov [1] proposed the elliptic stable envelope in equivariant elliptic cohomology, as a geometric framework to obtain elliptic R-matrices. This was made explicit [18] for cotangent bundles of Grassmannians, resulting in tensor products of two-dimensional irreducible representations of E τ, (sl 2 ). The higher rank case of sl N was studied later by H. Konno [46].Meanwhile, Nekrasov-Pestun-Shatashvili [49] from the 6d quiver gauge theory predicted the elliptic quantum group associated to an arbitrary Kac-Moody algebra, the precise definition of which (as an associative algebra) was proposed by . See also [52] in the context of quiver geometry.We are interested in the representation theory of E τ, (g) with ∈ C generic. The formal twist constructions [11,40] from U (Lg) might reduce the problem to the representation theory of affine quantum groups, which is a subject developed intensively in the last three decades from algebraic, geometric and combinatorial aspects. However loc.cit. involve formal power series of and infinite products in the comultiplication of E τ, (g). Some of these divergence issues was addressed [12] by Etingof-Moura, who defined a fully faithful tenor functor between representation categories of BGG type for U (Lsl N ) and E τ, (sl N ). Towards this functor not much is known: its image, the induced homomorphism of Grothendieck rings, etc.In this paper we study representations of E τ, (sl N ) via the RLL presentation [17] so as to bypass affine quantum groups, yet along the way we borrow ideas from the affine case. Compared to other works [7,12,19,32,44,45,51,52], our approach emphasizes more on the Grothendieck ring structure of representation category. It is a higher rank extension of a recent joint work with G. Felder [21].The presence of the dynamical parameter λ is one of the technical difficulties of elliptic quantum groups. ...

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Isomorphism Between the R-Matrix and Drinfeld Presentations of Yangian in Types B, C and D

Cited by 43 publications

References 19 publications

Spinorial R operator and Algebraic Bethe Ansatz

Spinorial R operator and Algebraic Bethe Ansatz

Isomorphism between the R-matrix and Drinfeld presentations of quantum affine algebra: Type C

Elliptic quantum groups and Baxter relations

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