I plan to discuss the constructions of different quantum groups . Our main tool - shuffle algebras . Definition of such algebras is rather elementary and simple.In shuffle terms it is possible to get the simple construction of affine and toroidal quantum groups and also in some cases deformed W-algebras. Representations of quantum groups and R -matrices are also naturally appear. Shuffle algebras are good for the constructions of coordinate rings of the quantizations of the different moduli spaces -for example instanton manifolds.I plan to explain how to do it.Actually such coordinate rings can be realised as quotients or subalgebras of the shuffle algebras. In some special case subalgebras in the suffle algebras become commutative. By this way we get "big" commutative subalgebras which in representations become the quantum hamiltonians of interesting integrable systems.

I plan to discuss the constructions of different quantum groups . Our main tool - shuffle algebras . Definition of such algebras is rather elementary and simple.In shuffle terms it is possible to get the simple construction of affine and toroidal quantum groups and also in some cases deformed W-algebras. Representations of quantum groups and R -matrices are also naturally appear. Shuffle algebras are good for the constructions of coordinate rings of the quantizations of the different moduli spaces -for example instanton manifolds.I plan to explain how to do it.Actually such coordinate rings can be realised as quotients or subalgebras of the shuffle algebras. In some special case subalgebras in the suffle algebras become commutative. By this way we get "big" commutative subalgebras which in representations become the quantum hamiltonians of interesting integrable systems.