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The deformed W-algebra is a quantum deformation of the W-algebra \(\mathcal{W}_\beta (\mathfrak {g})\) 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 \(\mathcal{W}_{x,r}\big (A_{2N}^{(2)}\big )\) , 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 \(\mathcal{W}_{x,r}\big (A(M,N)^{(1)}\big )\) via the quantum toroidal algebra.

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Quadratic Relations of the Deformed W-Algebra

  • 会议论文

  • 发表日期：

    2025年

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  • Takeo Kojima

摘要

The deformed W-algebra is a quantum deformation of the W-algebra \(\mathcal{W}_\beta (\mathfrak {g})\) 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 \(\mathcal{W}_{x,r}\big (A_{2N}^{(2)}\big )\) , 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 \(\mathcal{W}_{x,r}\big (A(M,N)^{(1)}\big )\) via the quantum toroidal algebra.