What is precisely a Yangian symmetry?

Question

The terms Yangian and Yangian symmetry appear in a list of physical problems (spin chains, Hubbard model, ABJM theory, $\mathcal{N}= 4$ super Yang-Mills in $d=4$, $\mathcal{N}= 8$ SUGRA in $d=4$), seem to be linked to (super)conformal symmetries and dual (super) conformal symmetries and Hopf algebras.

So, what is precisely a Yangian symmetry, and what is the physical signification of this symmetry?

This post imported from StackExchange Physics at 2015-06-15 19:41 (UTC), posted by SE-user Trimok

Answer 1

The Yangian is a deformation of the universal enveloping algebra of a certain Lie Algebra, whose generators satisfy the Yang-Baxter relation. For certain systems (such as those you mentioned) the generators commute with the Hamiltonian and as such the entire Yangian Hopf algebra constitutes symmetries of the system. The physical significance of these generators is that since they are represented by Hermitian (almost) local (that is, local in the thermodynamic limit) operators, related to currents, they are directly related to physical observables and corresponding conserved charges. Because, if you think about it, local Hermitian operators are physics.

This post imported from StackExchange Physics at 2015-06-15 19:41 (UTC), posted by SE-user Bubble

Answer 2

I found this, it may be of some help to answering your question:

An Introduction to Yangian Symmetries. Denis Bernard. Int. J. Mod. Phys. B 7, pp. 3517-3530 (1993). arXiv:hep-th/9211133.

This post imported from StackExchange Physics at 2015-06-15 19:41 (UTC), posted by SE-user alejandro123

Comments

Thanks, I will read it.

This post imported from StackExchange Physics at 2015-06-15 19:41 (UTC), posted by SE-user Trimok

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+1 This seems like a very good introduction.

This post imported from StackExchange Physics at 2015-06-15 19:41 (UTC), posted by SE-user Bubble