Quadratic relations of the deformed $W$-superalgebra ${\mathcal{W}}_{q,t}(A(M,N))$

Abstract

We find the free field construction of the basic $W$-current and screening currents for the deformed $W$-superalgebra ${\mathcal{W}}_{q,t}(A(M,N))$ associated with Lie superalgebra of type $A(M,N)$. Using this free field construction, we introduce the 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 Lie superalgebra $A(M,N)$, though the screening currents are not. This allows us to define ${\mathcal{W}}_{q,t}(A(M,N))$ by generators and relations.

arXiv comment28 pages, 35 figures. arXiv admin note: substantial text overlap with arXiv:1912.03096

References in corpus (12)

• Quantum Affine Algebras and Deformations of the Virasoro and W-algebras
• A Quantum Deformation of the Virasoro Algebra and the Macdonald Symmetric Functions
• Quantum W-algebras and Elliptic Algebras
• Quantum $W_N$ Algebras and Macdonald Polynomials
• Deformations of W-algebras associated to simple Lie algebras
• Drinfeld-Sokolov reduction for difference operators and deformations of W-algebras I. The case of Virasoro algebra
• Angular quantization and form-factors in massive integrable models
• Free Field Approach to the Dilute A_L Models
• $q$-Deformation of Corner Vertex Operator Algebras by Miura Transformation
• Quadratic relations of the deformed $W$-superalgebra ${\mathcal{W}}_{q,t}(\mathfrak{sl}(2|1))$
• The Integrals of Motion for the Deformed W-Algebra $W_{qt}(s{l}_N^{)}$ II: Proof of the commutation relations
• Comments on the Deformed W_N Algebra

Cited by in corpus (7)

• Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations
• A Note on Quiver Quantum Toroidal Algebra
• Quantum toroidal algebras and solvable structures in gauge/string theory
• 5d AGT correspondence of supergroup gauge theories from quantum toroidal ${\mathfrak{gl}}_1$
• Gauge origami and quiver W-algebras
• Gauge origami and quiver W-algebras III: Donaldson--Thomas $qq$-characters
• Quadratic Relations of the Deformed $W$-Algebra for the Twisted Affine Lie Algebra of Type $A_{2N}^{(2)}$