Quadratic relations of the deformed $W$-superalgebra ${\mathcal{W}}_{q,t}(\mathfrak{sl}(2|1))$

Abstract

We revisit the free field construction of the deformed $W$-superalgebras ${\mathcal{W}}_{q,t}(\mathfrak{sl}(2|1))$ by J. Ding and B. Feigin, {\it Contemp.Math.}{\bf 248}, 83-108 (1998), where the basic $W$-current and screening currents have been found. In this paper we introduce higher $W$-currents and obtain a closed set of quadratic relations among them. These relations are independent of the choice of Dynkin diagrams for the superalgebra $\mathfrak{sl}(2|1)$, though the screening currents are not. This allows us to define ${\mathcal{W}}_{q,t}(\mathfrak{sl}(2|1))$ by generators and relations.

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• Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations
• A Note on Quiver Quantum Toroidal Algebra
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• Quadratic relations of the deformed $W$-superalgebra ${\mathcal{W}}_{q,t}(A(M,N))$
• Gauge origami and quiver W-algebras
• Gauge origami and quiver W-algebras III: Donaldson--Thomas $qq$-characters
• Quantum deformation of Feigin-Semikhatov's W-algebras and 5d AGT correspondence with a simple surface operator
• Quadratic Relations of the Deformed $W$-Algebra for the Twisted Affine Lie Algebra of Type $A_{2N}^{(2)}$