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arXiv · 2312.16856

Quadratic relations of the deformed $W$-algebra

Takeo Kojima
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Abstract

The deformed $W$-algebra is a quantum deformation of the $W$-algebra ${\cal W}_β(\mathfrak{g})$ in conformal field theory. Using the free field construction, we obtain a closed set of quadratic relations of the $W$-currents of the deformed $W$-algebra. This allows us to define the deformed $W$-algebra by generators and relations. In this review, we study two types of deformed $W$-algebra. One is the deformed $W$-algebra ${\cal W}_{x,r}\big(A_{2N}^{(2)}\big)$, and the other is the $q$-deformed corner vertex algebra $q$-$Y_{L_1, L_2, L_3}$ that is a generalization of the deformed $W$-algebra ${\cal W}_{x,r}\big(A(M,N)^{(1)}\big)$ via the quantum toroidal algebra.

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Takeo Kojima. 2023-12-28. Quadratic relations of the deformed $W$-algebra. https://arxiv.org/abs/2312.16856

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