# Yangians and their applications

@inproceedings{Molev2002YangiansAT, title={Yangians and their applications}, author={Alexander I. Molev}, year={2002} }

Publisher Summary This chapter discusses the Yangians theory and their applications. The discovery of the Yangians is motivated by quantum inverse scattering theory. The Yangians form a remarkable family of quantum groups related to rational solutions of the classical Yang–Baxter equation. For each simple finite-dimensional Lie algebra α over the field of complex numbers, the corresponding Yangian Y (α) is defined as a canonical deformation of the universal enveloping algebra U (α[ x ]) for the… Expand

#### 95 Citations

Coideal subalgebras in quantum affine algebras

- Mathematics
- 2002

We introduce two subalgebras in the type A quantum affine algebra which are coideals with respect to the Hopf algebra structure. In the classical limit q -> 1 each subalgebra specializes to the… Expand

Yangians of Lie (super)algebras

- Mathematics
- 2006

This thesis is concerned with extending some well-known results about the Yangians Y (glN) and Y (slN) to the case of super-Yangians. First we produce a new presentation of the Yangian Y (glm|n),… Expand

On reflection algebras and twisted Yangians

- Physics, Mathematics
- 2004

It is well known that integrable models associated to rational R matrices give rise to certain non-Abelian symmetries known as Yangians. Analogously boundary symmetries arise when general but still… Expand

YANGIANS IN INTEGRABLE FIELD THEORIES, SPIN CHAINS AND GAUGE-STRING DUALITIES

- Mathematics, Physics
- 2012

In the following paper, which is based on the author's PhD thesis submitted to Imperial College London, we explore the applicability of Yangian symmetry to various integrable models, in particular,… Expand

Quantum Algebras and Integrable Boundaries in AdS/CFT

- Mathematics
- 2012

This thesis studies quantum integrable structures such as Yangians and quantum affine algebras that arise in and are inspired by the AdS/CFT duality, with a primary emphasis on the exploration of… Expand

L∞ - derivations and the argument shift method for deformation quantization algebras

- Mathematics
- 2019

The argument shift method is a well-known method for generating commutative families of functions in Poisson algebras from central elements and a vector field, verifying a special condition with… Expand

Casimir Elements for Some Graded Lie Algebras and Superalgebras

- Mathematics
- 2004

We consider a class of Lie algebras L such that L admits a grading by a finite Abelian group so that each nontrivial homogeneous component is one-dimensional. In particular, this class contains… Expand

KZ equation, G-opers, quantum Drinfeld–Sokolov reduction, and quantum Cayley–Hamilton identity

- Mathematics, Physics
- 2006

The Lax operator of Gaudin-type models is a 1-form at the classical level. In virtue of the quantization scheme proposed by D. Talalaev, it is natural to treat the quantum Lax operator as a… Expand

Rational Lax operators and their quantization

- Physics, Mathematics
- 2004

We investigate the construction of the quantum commuting hamiltonians for the Gaudin integrable model. We prove that [Tr L^k(z), Tr L^m(u) ]=0, for k,m < 4 . However this naive receipt of… Expand

KZ equation, G-opers and quantum Drinfeld-Sokolov reduction

- Physics
- 2006

The Lax operator of the classical Gaudin type models is a 1-form on the classical level. In virtue of the quantization scheme proposed in [Talalaev04] (hep-th/0404153) it is natural to treat the… Expand

#### References

SHOWING 1-10 OF 163 REFERENCES

Fundamental representations of Yangians and singularities of R-matrices.

- Mathematics
- 1991

The construction of Ä-matrices is closely related to the representation theory of quantum groups. In particular, there is an important class of quantum groups, called Yangians, such that to every… Expand

Finite dimensional irreducible representations of twisted Yangians

- Mathematics, Physics
- 1998

We study quantized enveloping algebras called twisted Yangians. They are analogs of the Yangian Y(gl(N)) for the classical Lie algebras of B, C, and D series. The twisted Yangians are subalgebras in… Expand

Yangians and Gelfand-Zetlin Bases

- Mathematics, Physics
- 1993

We establish a connection between the modern theory of Yangians and the classical construction of the Gelfand-Zetlin bases for the Lie algebra gln. Our approach allows us to produce the g-analogues… Expand

Yangians and transvector algebras

- Computer Science, Mathematics
- Discret. Math.
- 2002

We give a quantum analog of Sylvester's theorem where numerical matrices are replaced with noncommutative matrices whose entries are generators of the Yangian for the general linear Lie algebra… Expand

On the Harishchandra homomorphism for infinite-dimensional Lie algebras

- Mathematics
- 1984

A natural generalization of the finite-dimensional semisimple Lie algebras to the infinite-dimensional case is the Kac-Moody Lie algebra G(A) or more generally, the contragredient Lie algebra… Expand

NEWTON'S FORMULA FOR GLN

- Mathematics
- 1998

This paper presents an explicit relation between the two sets which are well-known generators of the center of the universal enveloping algebra U(gln) of the Lie algebra gln: one by Capelli (1890)… Expand

Yangians, integrable quantum systems and Dorey's rule

- Mathematics, Physics
- 1996

It was pointed out by P. Dorey that the three-point couplings between the quantum particles in affine Toda field theories have a remarkable Lie-theoretic interpretation. It is also well known that… Expand

A guide to quantum groups

- Mathematics
- 1994

Introduction 1. Poisson-Lie groups and Lie bialgebras 2. Coboundary Poisson-Lie groups and the classical Yang-Baxter equation 3. Solutions of the classical Yang-Baxter equation 4. Quasitriangular… Expand

Sklyanin determinant, Laplace operators, and characteristic identities for classical Lie algebras

- Mathematics
- 1995

The structure of quantized enveloping algebras called twisted Yangians, which are naturally associated with the B, C, and D series of the classical Lie algebras is studied. An explicit formula for… Expand

Multiparametric and Colored Extensions of the Quantum Group GLq(N) and the Yangian Algebra Y(glN) Through a Symmetry Transformation of the Yang–Baxter Equation

- Mathematics, Physics
- 1995

Inspired by Reshetikhin's twisting procedure to obtain multiparametric extensions of a Hopf algebra, a general "symmetry transformation" of the "particle conserving" R-matrix is found such that the… Expand