I have been learning about (classical) integrable systems lately, e.g. in the examples of a Lax pair etc. I frequently run into the term 'quantum integrable system'. May I ask a few questions:


1. What are quantum integrable systems? Are there examples that are not too complicated?
2. Why are mathematicians interested in quantum integrable systems?


I would appreciate any suitable references or papers. Thanks!



basic questions on quantum integrable systems
To answer the question *What are quantum integrable systems ?* let us find common understanding what is quantization. Roughly speaking it is the following - you have some classical phase space e.g. $\mathbb{R}^{2n}$ with coordinates $q\_i$ and $p\_i$. Now corresponding **quantum algebra** of observables is an algebra with generators $\hat p\_i, \hat q\_i$ such that commutators $[\hat p\_i , \hat q\_j]=\delta\_{ij}$.


Now assume you have classical **hamiltonian** integrable system, so you have functions $H\_i(p,q)$ such that they Poisson commute.


The first aim of quantization of integrable system is to find elements $\hat H\_i( \hat p , \hat q)$,
such that their commutator will be equal to zero and their principal symbols equals to classical $H\_i$.


**Answer 1a:** 
So a quantum integrable system corresponding to classical hamiltonian system defined by $H\_i$ is
given by $\hat H\_i$, with properties above. 


In general you may not consider $\mathbb{R}^{2n}$, but some symplectic manifold, and deformation quantization of algebra of functions on it. Quantization of integrable system - is again looking for $\hat H\_i$ in quantum algebra corresponding to $H\_i$ in algebra of functions on symplectic manifold.


Remark: Well, may you consider Poisson manifolds as well, and classical integrable systems are maximal Poisson commutative subalgebras and we may look for they lift to quantum algebras. Actually many practical examples work with Poisson manifolds g^\*, for Lie algebras $\frak g$.


To the best of my knowledge the question: `Is it always possible to do so or not?', is open in general. 
And in many case it is an **art** to find corresponding $H\_i$. Some recent research on general question can be found [here](http://arxiv.org/abs/1210.2840). Analysis of many concrete examples is related to representation theory, quantum groups etc...


**Answer 1b:** Are there examples that are not too complicated?


Trivial example - take $\mathbb R^{2n}$, take $H\_i = p\_i$, (free motion) corresponding quantum system $\hat H\_i = \hat p\_i$.


One more trivial example $H\_l=p\_l^2 + q\_l^2$ (independent Harmonic oscillators), corresponding $\hat H\_i = \hat p\_i ^2 + \hat q\_i^2$.


Let me give non-trivial examples later - you can search for Calogero system, Toda, Gaudin model.


**Answer 2:**  

Why are mathematicians interested in quantum integrable systems? 


1) In many cases study of quantum integrable systems is the same as certain representation theory questions - e.g. corresponding $\hat H\_i$ often have a sense of centres of universal enveloping or their quantum deformations [MO question about the Yangian](https://mathoverflow.net/questions/105399/question-about-the-yangian/105460#105460)  


2) It is related to certain "hot topics" as Langlands correspondence (e.g. <http://arxiv.org/abs/hep-th/0604128> ), quantum cohomology ( e.g. <http://arxiv.org/abs/1211.1287> )


3) We might want to apply math. to something "physically" sounding :)


