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Title:
Integrable Systems Related to Quantum Toroidal Algebras
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Abstract:
We give an introduction to the quantum toroidal algebras and their representation theory. Then we study a family of remarkable commutative subalgebras given explicitly by multiple integrals with elliptic kernels. We discuss the spectrum of these subalgebras in various modules and relations to more familiar integrable systems.Our main examples are quantum toroidal algebras associated to ${\mathfrak gl}(n)$ and quantum comodule algebras ${\mathcal K}_n$ with emphasis on the case of $n=1$.
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