Abstracts

Takeo Kojima

Integrable Structure of deformed W-algebra $W_{q,t}({\mathfrak g})$, quadratic relations and integrals of motion

Deformed W-algebra $W_{q,t}({\mathfrak g})$ is an elliptic deformation (a two-parameter deformation) of the classical W-algebra ${\cal W}({\mathfrak g})$ in the KdV theory. We study integrable structure of deformed W-algebra $W_{q,t}({\mathfrak g})$. This talk is divided into Part I: quadratic relations and Part II: integrals of motion. In Part I, we obtain a closed set of quadratic relations of the W currents of the deformed W algebra $W_{q,t}({\mathfrak g})$. This allows us to define the deformed W algebra by generators and relations. We report the case ${\mathfrak g}=A(M,N)^{(1)}, A_{2N}^{(2)}$. In Part II, we study an elliptic deformation of the ${\mathfrak g}$-KdV theory. We construct infinitely many commutative operators G_n that we call the non-local integrals of motion for $W_{q,t}({\mathfrak g})$. These operators G_n can be regarded as an elliptic deformation of the Hamiltonian of the ${\mathfrak g}$-KdV theory. We report the case ${\mathfrak g}=D_N^{(1)}$. We start this talk with basic reviews on the W algebra in the conformal field theory, so that even beginners easily understand it without much prior knowledge. Part II is based on collaboration with Professor Michio Jimbo.