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We study a W-algebra of central charge 2(k-1)/(k+2) with k a positive integer greater than 1
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It is proved that the parafermion vertex operator algebra associated to the irreducible highest weight module for the affine Kac-Moody algebra A_1^{(1)} of level k coincides with a certain W-algebra. In particular, a set of generators for the parafermion vertex operator algebra is determined.
We discuss some aspects of the deformed W-algebras W_{q,t}[g]. In particular, we derive an explicit formula for the Kac determinant, and discuss the center when t^2 is a primitive k-th root of unity. The relation of the structure of W_{q,t}[g] to the representation ring of the quantum affine algebra U_q(hat g), as discovered recently by Frenkel and Reshetikhin, is further elucidated in some examples.
The deformed $mathcal W$ algebras of type $textsf{A}$ 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 $textsf{B},textsf{C},textsf{D}$ 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 $textsf{D}^{(2)}_{ell+1}$. 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.
The structure of the parafermion vertex operator algebra associated to an integrable highest weight module for any affine Kac-Moody algebra is studied. In particular, a set of generators for this algebra has been determined.
The trace (or zeroth Hochschild homology) of Khovanovs Heisenberg category is identified with a quotient of the algebra W_{1+infty}. This induces an action of W_{1+infty} on symmetric functions.
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