Twisted Yangians for symmetric pairs of types B, C, D

Guay, Nicolas; Regelskis, Vidas

doi:10.1007/s00209-016-1649-2

Cited by 18 publications

(24 citation statements)

References 40 publications

(74 reference statements)

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“…By [GR,Thm. 3.1] the algebra Y (g N , G) tw is isomorphic to the subalgebra of Y (g N ) generated by the coefficients σ (r) ij with r ≥ 0 of the matrix entries σ ij (u) of the S-matrix Σ(u) defined by (3.16)…”

Section: Symmetric Pairs Of Classical Lie Algebrasmentioning

confidence: 99%

“…Moreover, c(u) satisfies the relation c(u) −1 = c(κ − u) and the odd coefficients are algebraically independent. By [GR,Thm. 4.2 and Cor.…”

Section: Symmetric Pairs Of Classical Lie Algebrasmentioning

confidence: 99%

“…Reflection algebras of types B-C-D. It was shown in [GR,Sec. 4] that certain quotients of the extended reflection algebra X B(G) defined by the reflection equation are isomorphic to twisted Yangians…”

Section: Symmetric Pairs Of Classical Lie Algebrasmentioning

confidence: 99%

“…This implies that Y (g N , G) tw , viewed as a subalgebra of X(g N ), is the ν g -stable subalgebra of X(g N , G) tw ; see [GR,Cor. 3.1].…”

Section: Symmetric Pairs Of Classical Lie Algebrasmentioning

confidence: 99%

“…It was noted in [GR,Remark 3.2], that the automorphism α A of X(g N ) (see (3.3)) restricts to an isomorphism between the algebras X(g N , G) tw and X(g N , A t GA) tw . Given a matrix G let A ∈ GL(N ) be such that AA t = I and A t GA = G. Then α A restricts to an automorphism of X(g N , G) tw .…”

Section: Symmetric Pairs Of Classical Lie Algebrasmentioning

confidence: 99%

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Representations of twisted Yangians of types B, C, D: I

Guay

Regelskis

Wendlandt

2017

Sel. Math. New Ser.

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Section: Symmetric Pairs Of Classical Lie Algebrasmentioning

confidence: 99%

“…Moreover, c(u) satisfies the relation c(u) −1 = c(κ − u) and the odd coefficients are algebraically independent. By [GR,Thm. 4.2 and Cor.…”

Section: Symmetric Pairs Of Classical Lie Algebrasmentioning

confidence: 99%

Section: Symmetric Pairs Of Classical Lie Algebrasmentioning

confidence: 99%

“…This implies that Y (g N , G) tw , viewed as a subalgebra of X(g N ), is the ν g -stable subalgebra of X(g N , G) tw ; see [GR,Cor. 3.1].…”

Section: Symmetric Pairs Of Classical Lie Algebrasmentioning

confidence: 99%

Section: Symmetric Pairs Of Classical Lie Algebrasmentioning

confidence: 99%

See 3 more Smart Citations

Representations of twisted Yangians of types B, C, D: I

Guay

Regelskis

Wendlandt

2017

Sel. Math. New Ser.

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Spin chain overlaps and the twisted Yangian

Leeuw

Gombor

Kristjansen³

et al. 2020

J. High Energ. Phys.

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Using considerations based on the thermodynamical Bethe ansatz as well representation theory of twisted Yangians we derive an exact expression for the overlaps between the Bethe eigenstates of the SO(6) spin chain and matrix product states built from matrices whose commutators generate an irreducible representation of so(5). The latter play the role of boundary states in a domain wall version of N = 4 SYM theory which has nonvanishing, SO(5) symmetric vacuum expectation values on one side of a co-dimension one wall. This theory, which constitutes a defect CFT, is known to be dual to a D3-D7 probe brane system. We likewise show that the same methodology makes it possible to prove an overlap formula, earlier presented without proof, which is of relevance for the similar D3-D5 probe brane system.

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Gauge theory and boundary integrability

Roland

Skinner

2019

J. High Energ. Phys.

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We study the mixed topological / holomorphic Chern-Simons theory of Costello, Witten and Yamazaki on an orbifold (Σ × C)/Z 2 , obtaining a description of lattice integrable systems in the presence of a boundary. By performing an order calculation we derive a formula for the the asymptotic behaviour of K-matrices associated to rational, quasi-classical R-matrices. The Z 2 -action on Σ × C fixes a line L, and line operators on L are shown to be labelled by representations of the twisted Yangian. The OPE of such a line operator with a Wilson line in the bulk is shown to give the coproduct of the twisted Yangian. We give the gauge theory realisation of the Sklyanin determinant and related conditions in the RT T presentation of the boundary Yang-Baxter equation. Introduction The Yang-Baxter EquationInteractions of an integrable spin chain are determined by an object known as an R-matrix. This is a linear mappair of complex vector spaces and z, z ′ complex spectral parameters on which the R-matrix depends meromorphically. The spin chain is integrable if the R-matrix obeys the Yang-Baxter equation (YBE)(1.1) around the identity 1 V ⊗V ′ . R-matrices admitting such an expansion are called quasiclassical, and r(z, z ′ ) is known as the classical r-matrix. This classical r-matrix takes values in g ⊗ g for g a finite dimensional, complex, simple Lie algebra, acting in representations associated to V ⊗ V ′ . Expanding the YBE to second order in shows that the classical r-matrix obeys the classical Yang-Baxter equation,Solutions of the classical Yang-Baxter equations were classified by Belavin & Drinfeld [4], under the (mild) assumption that the r-matrix is non-degenerate. The solutions can be separated into three families, distinguished by whether the classical r-matrix can be written in terms of rational, trigonometric, or elliptic functions. In this work we will concentrate on the rational case. Each family of solutions to is associated with an algebra, which in the rational case is the Yangian Y(g). Indeed, in [5] it was demonstrated that all rational solutions of the YBE of the form (1.1) determine a representation of the Yangian on V , and are themselves determined by such a representation. Accordingly, the line operators in Costello's theory are actually associated with representations of Y(g). These include ordinary Wilson lines in certain representations of g itself, but also more general line operators. As shown in [1-3], the more general line operators arise even in the OPE of two ordinary Wilson lines.Since the YBE is homogeneous, its solutions are defined only up to multiplication by a function f (z 1 , z 2 ) of the spectral parameters. This degeneracy can be removed by required that the R-matrix obeys a constraint known as the quantum determinant condition. In the rational case, for V the defining vector representation of g = sl n (C), the quantum determinant [6] condition is enough to guarantee existence of a unique quasi-classical Rmatrix in this representation for a given classical r-matrix. Similar results e...

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Twisted Yangians for symmetric pairs of types B, C, D

Cited by 18 publications

References 40 publications

Representations of twisted Yangians of types B, C, D: I

Representations of twisted Yangians of types B, C, D: I

Spin chain overlaps and the twisted Yangian

Gauge theory and boundary integrability

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