# Yangian

A **Yangian** is a [Hopf algebra](https://www.edgepedia.org/hopf-algebra) (an algebraic structure with multiplication plus compatible coproduct operations), introduced by Vladimir G. Drinfeld in 1985, that forms an infinite-dimensional deformation of the universal enveloping algebra of the current algebra of a finite-dimensional simple [Lie algebra](https://www.edgepedia.org/lie-algebra) and supplies the algebraic structure behind the quantum Yang-Baxter equation.<sup>[1](https://www.mathnet.ru/eng/dan/v283/i5/p1060)</sup><sup> • </sup><sup>[2](https://researchers.usask.ca/curtis-wendlandt/documents/r-matrix-yangian.pdf)</sup><sup> • </sup><sup>[20](https://arxiv.org/abs/2009.06934)</sup> Together with the quantum affine algebras, which deform the same enveloping algebras in a trigonometric way, Yangians are the two principal families of quantum groups; the Yangian is distinguished by its rational dependence on the spectral parameter.<sup>[3](https://rxiv.org/pdf/1111.0091v1.pdf)</sup> In physics, the Yangian organizes the conserved charges of integrable systems, including two-dimensional field theories, spin chains, and the planar [AdS/CFT correspondence](https://www.edgepedia.org/ads-cft-correspondence).<sup>[4](https://arxiv.org/abs/1201.1884)</sup>

| Key fact | Detail |
|---|---|
| Origin | Introduced by V. G. Drinfeld, "Hopf algebras and the quantum Yang-Baxter equation", Dokl. Akad. Nauk SSSR 283:5 (1985), pp. 1060-1064, written in Russian<sup>[1](https://www.mathnet.ru/eng/dan/v283/i5/p1060)</sup> |
| Algebraic type | A Hopf algebra with coproduct, counit, and antipode; its universal R-matrix satisfies the quantum Yang-Baxter equation<sup>[5](https://citeseerx.ist.psu.edu/document?doi=c77c0ea62624f193368058323a1df903947b3cba&repid=rep1&type=pdf)</sup> |
| Presentations | At least three: Drinfeld's original ("J") presentation, the R-matrix (RTT) realization, and the Drinfeld "new" (current) presentation<sup>[2](https://researchers.usask.ca/curtis-wendlandt/documents/r-matrix-yangian.pdf)</sup> |
| Rational R-matrix | For the Yangian the R-matrix is rational in the spectral parameter, while for the quantum affine algebra it is trigonometric<sup>[3](https://rxiv.org/pdf/1111.0091v1.pdf)</sup> |
| Representation theory | Finite-dimensional irreducible representations are classified by polynomials, the Drinfeld polynomials<sup>[6](https://uhra.herts.ac.uk/id/eprint/7393/2/VR_AAM.pdf)</sup> |
| Physical role | Unifies the symmetries constraining S-matrices of integrable models; scattering matrices can be reconstructed directly from the Yangian<sup>[4](https://arxiv.org/abs/1201.1884)</sup> |
| Classical limit | The Yangian also appears in classical integrable field theory as a Poisson-Hopf algebra<sup>[7](https://www.sciencedirect.com/science/article/pii/037026939290280H)</sup> |

## Definition and structure

Drinfeld's 1985 paper addressed a question raised by the [quantum inverse scattering method](https://www.edgepedia.org/quantum-inverse-scattering-method): given a finite-dimensional simple Lie algebra over \( \mathbb{C} \) with a fixed invariant inner product, and a representation \( \rho: \mathfrak{a} \to \mathrm{End}\,V \), does the quantum Yang-Baxter equation admit a solution expressible as a formal series? The Yangian \( Y(\mathfrak{g}) \) was formalized in this pioneering paper, where several foundational results were established.<sup>[1](https://www.mathnet.ru/eng/dan/v283/i5/p1060)</sup><sup> • </sup><sup>[2](https://researchers.usask.ca/curtis-wendlandt/documents/r-matrix-yangian.pdf)</sup> Drinfeld's broader 1986 survey "Quantum groups" placed the construction in the general Hopf-algebra formalism designed to serve as the algebraic foundation of the quantum inverse scattering method.<sup>[8](https://m.mathnet.ru/php/archive.phtml?jrnid=znsl&option_lang=eng&paperid=5159&wshow=paper)</sup>

As a Hopf algebra, the Yangian \( Y(\mathfrak{g}) \) carries a coproduct \( \Delta: Y \to Y \otimes Y \), a counit, and an antipode. The coproduct acts on the L-operator as \( (\mathrm{id} \otimes \Delta) L(\lambda) = L_{13}(\lambda)\, L_{12}(\lambda) \), and it is this structure that lets single-site objects be extended to many-body systems.<sup>[5](https://citeseerx.ist.psu.edu/document?doi=c77c0ea62624f193368058323a1df903947b3cba&repid=rep1&type=pdf)</sup> The algebra also possesses a universal R-matrix, a series \( R(u) \) that itself satisfies the quantum Yang-Baxter equation.<sup>[9](https://www.dissercat.com/content/spektry-podalgebr-bete-v-yangianakh)</sup>

## Presentations

Yangians admit at least three important presentations: Drinfeld's original (or "J") presentation, the R-matrix (or RTT) realization, and the Drinfeld "new" (or current) presentation.<sup>[2](https://researchers.usask.ca/curtis-wendlandt/documents/r-matrix-yangian.pdf)</sup> For the Yangian \( Y_n = Y(\mathfrak{gl}_n) \), the RTT formalism starts from Yang's R-matrix,

\[ R(u) = u - \sum_{i,j=1}^{n} e_{i,j} \otimes e_{j,i} \in M_n \otimes M_n[u], \]

which satisfies the quantum Yang-Baxter equation with spectral parameters,

\[ R^{[1,2]}(u-v)\, R^{[1,3]}(u)\, R^{[2,3]}(v) = R^{[2,3]}(v)\, R^{[1,3]}(u)\, R^{[1,2]}(u-v). \]

The algebra \( Y_n \) is then generated by elements \( T^{(r)}_{i,j} \) for \( 1 \le i,j \le n \) and \( r \ge 1 \), collected into a formal series \( T_{i,j}(u) = \sum_{r \ge 0} T^{(r)}_{i,j} u^{-r} \), subject to relations written in terms of this R-matrix.<sup>[10](https://pages.uoregon.edu/brundan/papers/drinfeldx.pdf)</sup> A complete proof of the equivalence of the RTT presentation of \( Y_n \) with Drinfeld's presentation filled a gap in the literature, and the parabolic presentation generalizes the RTT presentation, which is recovered in the special case \( \nu = (n) \).<sup>[10](https://pages.uoregon.edu/brundan/papers/drinfeldx.pdf)</sup><sup> • </sup><sup>[11](https://darkwing.uoregon.edu/~brundan/papers/drinfeldx.pdf)</sup> For \( \mathfrak{gl}_n \), the classification of finite-dimensional irreducible representations can be proved in either the RTT or the current presentation, and the two families of classifying polynomials are related through the Gauss decomposition.<sup>[6](https://uhra.herts.ac.uk/id/eprint/7393/2/VR_AAM.pdf)</sup> Isomorphisms between the R-matrix and Drinfeld presentations, established via Gauss decomposition, also underlie the description of finite-dimensional irreducible representations for the Yangians of types B, C, and D.<sup>[12](https://extern.maths.usyd.edu.au/u/pubs/publist/preprints/2020/jing-21.html)</sup>

## Representation theory

It has been known since Drinfeld's work that finite-dimensional irreducible representations of Yangians are classified by certain polynomials, usually called Drinfeld polynomials; Drinfeld proved the classification using the current presentation.<sup>[6](https://uhra.herts.ac.uk/id/eprint/7393/2/VR_AAM.pdf)</sup> A particularly useful representation is the evaluation representation: given a representation \( \rho \) of the Lie algebra \( \mathfrak{g} \), one sets \( \rho(\hat{J}^A) = u\, \rho(J^A) \), where \( u \) is a spectral parameter, though this identification does not work for all representations \( \rho \).<sup>[13](https://edu.itp.phys.ethz.ch/fs13/int/SpinChains.pdf)</sup> Similar classification problems arise for super analogs: for the super Yangian \( Y(\mathfrak{gl}(M,N)) \), Zhang built a highest weight framework for its finite-dimensional representations and used it to classify the finite-dimensional simple modules.<sup>[14](https://link.springer.com/article/10.1007/s00209-014-1330-6)</sup>

## Classical origins

The Yangian is not only a quantum object. In 1992, N. J. MacKay demonstrated that Drinfeld's Yangian algebra—recently examined as the algebra formed by conserved charges in some two-dimensional integrable quantum field theories—likewise appears in the classical setting in the form of a [Poisson-Hopf algebra](https://www.edgepedia.org/lie-bialgebra), and he set out the Serre relations, coproduct, and antipode in explicit form.<sup>[7](https://www.sciencedirect.com/science/article/pii/037026939290280H)</sup> In the same setting of principal chiral models in 1+1 dimensions, a classically conserved local charge exists for each totally symmetric invariant tensor of the underlying group, and these local charges are in involution with the non-local Yangian charges.<sup>[7](https://www.sciencedirect.com/science/article/pii/037026939290280H)</sup>

## Role in integrable systems

**Spin chains.** In the Heisenberg spin chain, expanding the monodromy matrix at large spectral parameter first yields the total spin operators, which form an \( \mathfrak{su}(2) \) algebra of local symmetries; the next order produces non-local terms, products of spin operators at two different sites, which are the first Yangian generators beyond the Lie algebra.<sup>[13](https://edu.itp.phys.ethz.ch/fs13/int/SpinChains.pdf)</sup> The commutator of two Yangian generators is again a generator, so the algebra is infinite dimensional, and the coproduct takes the form \( \Delta \hat{J}^A = \hat{J}^A \otimes 1 + 1 \otimes \hat{J}^A + f^A_{\;BC}\, J^B \otimes J^C \).<sup>[13](https://edu.itp.phys.ethz.ch/fs13/int/SpinChains.pdf)</sup> The coproduct is not invariant under permutations, and this asymmetry has a concrete consequence: requiring Yangian invariance of the R-matrix, with the evaluation representation supplying the two spectral parameters \( u_1 \) and \( u_2 \), fixes the remaining coefficient uniquely and yields \( R = (u_1 - u_2)\mathbf{1} + iP \), so the R-matrix and the entire model are fixed by the symmetries alone.<sup>[13](https://edu.itp.phys.ethz.ch/fs13/int/SpinChains.pdf)</sup> More generally, the monodromy matrix built from \( N \) copies of the Yangian via the iterated coproduct yields a transfer matrix \( t(\lambda) = \mathrm{Tr}_0\, T_0(\lambda) \) whose elements form a family of commuting operators, \( [t(\lambda), t(\lambda')] = 0 \), which is what ensures integrability of the periodic spin chain.<sup>[5](https://citeseerx.ist.psu.edu/document?doi=c77c0ea62624f193368058323a1df903947b3cba&repid=rep1&type=pdf)</sup>

**Scattering matrices and field theory.** It had long been known that S-matrices in integrable models are fixed by symmetry, but Lie algebra symmetry, the Yang-Baxter equation, crossing, and unitarity were usually treated as separate, independent properties. In a 2012 review drawn from his [Imperial College London](https://www.edgepedia.org/imperial-college-london) dissertation, Fabian Spill demonstrated that scattering matrices can be built from the Yangian alone, establishing it as the algebraic structure uniting all symmetries needed by many integrable models; starting from the Yangian, he recovered the principal chiral field's S-matrix and, aside from a CDD factor, that of further integrable field theories possessing \( \mathfrak{su}(n) \) symmetry.<sup>[4](https://arxiv.org/abs/1201.1884)</sup>

**Planar gauge theories and AdS/CFT.** The planar limit of \( \mathcal{N}=4 \) supersymmetric Yang-Mills theory is believed to be integrable, and Yangian symmetry appears throughout. The S-matrix describing scattering of magnon excitations of the associated spin chain has exact Yangian symmetry even at finite coupling strength, though this symmetry is based on a subalgebra of, or more precisely an extension of, the expected algebra; color-ordered scattering amplitudes of ordinary particles of the four-dimensional gauge theory also display Yangian invariance, understood as an interplay between ordinary and dual conformal symmetry.<sup>[15](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.98.046006)</sup> Classical Yangian symmetry has been established for the action of planar \( \mathcal{N}=4 \) super Yang-Mills and related planar gauge theories, with the bilocal generators acting on the cyclic action so that it becomes exactly invariant.<sup>[15](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.98.046006)</sup> Tree-level amplitudes in this theory have also been approached through integrable spin chains, Yangian invariance, and [Grassmannian](https://www.edgepedia.org/grassmannian) contour integrals.<sup>[16](https://epub.ub.uni-muenchen.de/82874/)</sup>

**QCD.** Yangian symmetry reaches phenomenology as well. In 2023, Roland Kirschner of the Institut für Theoretische Physik, Universität Leipzig, surveyed applications in high-energy QCD, presenting Yangian symmetric correlators built within the Yangian algebra of type \( g\ell(n) \) and using them for tree-level scattering amplitudes and their iterative structure, for amplitudes of parton splitting together with the kernels in the scale-evolution equations of structure functions, and for the equations governing the asymptotic high-energy behavior of scattering.<sup>[17](https://arxiv.org/abs/2302.00449)</sup>

## Variants and related algebras

The Yangian sits in a family of Hopf-algebraic structures built around the Yang-Baxter equation. Twisted Yangians arise in the study of reflection algebras, where the coproduct structure of the ordinary Yangian is used to build the monodromy matrix of a periodic spin chain and its transfer matrix.<sup>[5](https://citeseerx.ist.psu.edu/document?doi=c77c0ea62624f193368058323a1df903947b3cba&repid=rep1&type=pdf)</sup> Reflection matrices related to Yangian and super-Yangian R-matrices associated with the infinite series \( \mathfrak{so}(m) \), \( \mathfrak{sp}(n) \), and \( \mathfrak{osp}(m|n) \) have been classified for open spin chain models.<sup>[18](https://arxiv.org/abs/math/0304150)</sup> Since the 1980s the study of Yangians has grown into a theory with applications to the theory of classical Lie algebras, finite W-algebras and their representations, classical W-algebras and affine vertex algebras, and geometric representation theory.<sup>[2](https://researchers.usask.ca/curtis-wendlandt/documents/r-matrix-yangian.pdf)</sup> Yangian-type presentations of the semiclassical limits of W-algebras attached to distinguished [nilpotent elements](https://www.edgepedia.org/nilpotent-element) in classical Lie algebras have been obtained using Dirac reduction.<sup>[19](https://paperity.org/p/309601291/one-dimensional-representations-of-finite-w-algebras-dirac-reduction-and-the-orbit-method)</sup>

## References

1. [V. G. Drinfeld, "Hopf algebras and the quantum Yang-Baxter equation", Dokl. Akad. Nauk SSSR 283:5 (1985) 1060-1064, Mathnet](https://www.mathnet.ru/eng/dan/v283/i5/p1060)
2. [C. Wendlandt, "The R-matrix presentation for the Yangian of a simple Lie algebra", University of Saskatchewan](https://researchers.usask.ca/curtis-wendlandt/documents/r-matrix-yangian.pdf)
3. ["About the structure of the Yangian algebra", rxiv.org preprint](https://rxiv.org/pdf/1111.0091v1.pdf)
4. [F. Spill, "Yangians in Integrable Field Theories, Spin Chains and Gauge-String Dualities", arXiv:1201.1884](https://arxiv.org/abs/1201.1884)
5. ["On reflection algebras and twisted Yangians", CiteSeerX](https://citeseerx.ist.psu.edu/document?doi=c77c0ea62624f193368058323a1df903947b3cba&repid=rep1&type=pdf)
6. ["Yangian of gl_n: RTT and current presentations", University of Hertfordshire](https://uhra.herts.ac.uk/id/eprint/7393/2/VR_AAM.pdf)
7. [N. J. MacKay, "On the classical origins of Yangian symmetry in integrable field theory", Physics Letters B 281 (1992) 90-97](https://www.sciencedirect.com/science/article/pii/037026939290280H)
8. [V. G. Drinfeld, "Quantum groups", Zap. Nauchn. Sem. LOMI 155 (1986) 18-49, Mathnet](https://m.mathnet.ru/php/archive.phtml?jrnid=znsl&option_lang=eng&paperid=5159&wshow=paper)
9. ["Спектры подалгебр Бете в Янгианах", dissercat](https://www.dissercat.com/content/spektry-podalgebr-bete-v-yangianakh)
10. [J. Brundan, "Presentations of the Yangian", University of Oregon](https://pages.uoregon.edu/brundan/papers/drinfeldx.pdf)
11. [J. Brundan, "Parabolic presentations of the Yangian Y(gl_n)", University of Oregon](https://darkwing.uoregon.edu/~brundan/papers/drinfeldx.pdf)
12. [N. Jing, M. Liu, A. Molev, "Representations of quantum affine algebras in their R-matrix realization", University of Sydney](https://extern.maths.usyd.edu.au/u/pubs/publist/preprints/2020/jing-21.html)
13. ["Spin chains", ETH Zurich lecture notes](https://edu.itp.phys.ethz.ch/fs13/int/SpinChains.pdf)
14. ["Representations of quantum affine superalgebras", Mathematische Zeitschrift, Springer](https://link.springer.com/article/10.1007/s00209-014-1330-6)
15. ["Yangian symmetry for the action of planar super Yang-Mills and super Chern-Simons theories", Phys. Rev. D 98, 046006 (2018)](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.98.046006)
16. ["Grassmannian integrals in Minkowski signature, amplitudes, and integrability", LMU Munich](https://epub.ub.uni-muenchen.de/82874/)
17. [R. Kirschner, "Yangian symmetry applied to Quantum chromodynamics", arXiv:2302.00449](https://arxiv.org/abs/2302.00449)
18. ["Classification of reflection matrices related to (super) Yangians and application to open spin chain models", arXiv:math/0304150](https://arxiv.org/abs/math/0304150)
19. ["One dimensional representations of finite W-algebras, Dirac reduction and the orbit method", Paperity](https://paperity.org/p/309601291/one-dimensional-representations-of-finite-w-algebras-dirac-reduction-and-the-orbit-method)
20. [On classical limits of Bethe subalgebras in Yangians](https://arxiv.org/abs/2009.06934)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Hopf and quantum algebras › Applications of Hopf and quantum algebras*

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