Mathematics

In this talk I will explain the definition of shuffle algebras. Construction of shuffle algebras is simple and general. As a special cases we get affine quantum groups, Sklyanin algebras, toroidal algebras and, in the limit, vertex operator algebras. Sometimes in the limit we have familiar conformal theories related with W-algebras. I will present several examples and applications. Most popular applications are based on the fact that toroidal-like algebras are acting in the K-theory of modular spaces by Nakajima-type correspondence.