nLab Yangian

Idea

A Yangian is a certain quantum group that arises naturally in integrable systems in quantum field theory, as well as in semi-holomorphic 4d Chern-Simons theory.

Related Concepts

• quantum affine algebra

References

• Wikipedia, Yangian

• A. I. Molev, Yangians and their applications, in “Handbook of Algebra” vol. 3 (M. Hazewinkel, Ed.), Elsevier 2003, 907-959 math.QA/0211288

• A. I. Molev, Yangians and classical Lie algebras, AMS Math. Surv. Monog. 143, 2007; 400 pp; Russian edition: Янгианы и классические алгебры Ли, МЦНМО, Москва, 2009

• N. J. Mackay, Introduction to Yangian symmetry in integrable field theory (arXiv:hep-th/0409183)

• Vassili Gorbounov, R. Rimanyi, V. Tarasov, A. Varchenko, Cohomology of the cotangent bundle of a flag variety as a Yangian Bethe algebra, arXiv:1204.5138

• V. G. Drinfeld, Degenerate affine Hecke algebras and Yangians, Funct. Anal. Appl. 20 (1986), 58–60.

• Denis Uglov, Symmetric functions and the Yangian decomposition of the Fock and basic modules of the affine Lie algebra $\mathfrak{sl}^N$, Math. Soc. Japan Memoirs 1, 1998, 183-241 euclid doi

• A. N. Kirillov, N. Y. Reshetikhin, The Yangians, Bethe Ansatz and combinatorics, Lett. Math. Phys. 12, 199 (1986)

• Sachin Gautam, Valerio Toledano-Laredo, Yangians and quantum loop algebras, Selecta Mathematica 19 (2013), 271-336 arXiv:1012.3687; Yangians, quantum loop algebras and abelian difference equations (Formerly: Yangians and quantum loop algebras II. Equivalence of categories via abelian difference equations) J. Amer. Math. Soc. 29 (2016) 775–824 arXiv:1310.7318; III. Meromorphic tensor equivalence for Yangians and quantum loop algebras, Publ.math. IHES 125, 267–337 (2017). doi arXiv:1403.5251

Braided Yangians and Yangians associated to R-matrices were defined and studied in

• Dmitri I. Gurevich, Pavel A. Saponov, Generalized Yangians and their Poisson counterparts, Theor. Math. Phys. 192, 1243–1257 (2017) doi

Yangians for quivers and relation to quantum equivariant cohomology of Nakajima’s quiver varieties

• Davesh Maulik, Andrei Okounkov, Quantum groups and quantum cohomology, Astérisque 408, SMF 2019 arXiv:1211.1287

Review in the context of AdS-CFT includes

• Alessandro Torrielli, Yangians, S-matrices and AdS/CFT, J.Phys.A44:263001,2011 (arXiv:1104.2474)

Quiver Yangians appearing in description of Hall algebras of $\omega$-semistable compactly supported sheaves with fixed slope on resolutions of Kleinian singularities:

• Duiliu-Emanuel Diaconescu, Mauro Porta, Francesco Sala, McKay correspondence, cohomological Hall algebras and categorification, arXiv:2004.13685

In

• Kevin Costello, Supersymmetric gauge theory and the Yangian (arXiv:1303.2632)

is discussed that the holomorphically twisted N=1 D=4 super Yang-Mills theory is controled by the Yangian in analogy to how Chern-Simons theory is controled by a quantum group. See at semi-holomorphic 4d Chern-Simons theory.