Deformations of $\mathcal{W}$ algebras via quantum toroidal algebras

Abstract

The deformed $\mathcal{W}$ algebras of type $\text{A}$ have a uniform description in terms of the quantum toroidal ${\mathfrak{gl}}_1$ algebra $\mathcal{E}$. We introduce a comodule algebra $\mathcal{K}$ over $\mathcal{E}$ which gives a uniform construction of basic deformed $\mathcal{W}$ currents and screening operators in types $\text{B},\text{C},\text{D}$ including twisted and supersymmetric cases. We show that a completion of algebra $\mathcal{K}$ contains three commutative subalgebras. In particular, it allows us to obtain a commutative family of integrals of motion associated with affine Dynkin diagrams of all non-exceptional types except ${\text{D}}_{\ell +1}^{(2)}$. We also obtain in a uniform way deformed finite and affine Cartan matrices in all classical types together with a number of new examples, and discuss the corresponding screening operators.

Latex 53 pages. Several misprints are corrected

Cited by in corpus (2)

• Quadratic relations of the deformed $W$-superalgebra ${\mathcal{W}}_{q,t}(\mathfrak{sl}(2|1))$
• Combinatorics of vertex operators and deformed $W$-algebra of type D$(2,1;\alpha )$