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Register a new userClassical $mathcal{W}$-algebras for centralizers
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We introduce a new family of Poisson vertex algebras $mathcal{W}(mathfrak{a})$ analogous to the classical $mathcal{W}$-algebras. The algebra $mathcal{W}(mathfrak{a})$ is associated with the centralizer $mathfrak{a}$ of an arbitrary nilpotent element in $mathfrak{gl}_N$. We show that $mathcal{W}(mathfrak{a})$ is an algebra of polynomials in infinitely many variables and produce its free generators in an explicit form. This implies that $mathcal{W}(mathfrak{a})$ is isomorphic to the center at the critical level of the affine vertex algebra associated with $mathfrak{a}$.
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We introduce a new family of affine $W$-algebras associated with the centralizers of arbitrary nilpotent elements in $mathfrak{gl}_N$. We define them by using a version of the BRST complex of the quantum Drinfeld--Sokolov reduction. A family of free generators of the new algebras is produced in an explicit form. We also give an analogue of the Fateev--Lukyanov realization for these algebras by applying a Miura-type map.
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.
We consider the centers of the affine vertex algebras at the critical level associated with simple Lie algebras. We derive new formulas for generators of the centers in the classical types. We also give a new formula for the Capelli-type determinant for the symplectic Lie algebras and calculate the Harish-Chandra images of the Casimir elements arising from the characteristic polynomial of the matrix of generators of each classical Lie algebra.
We investigate the irreducibility of the nilpotent Slodowy slices that appear as the associated variety of W-algebras. Furthermore, we provide new examples of vertex algebras whose associated variety has finitely many symplectic leaves.
For every simple Lie algebra $mathfrak{g}$ we consider the associated Takiff algebra $mathfrak{g}^{}_{ell}$ defined as the truncated polynomial current Lie algebra with coefficients in $mathfrak{g}$. We use a matrix presentation of $mathfrak{g}^{}_{ell}$ to give a uniform construction of algebraically independent generators of the center of the universal enveloping algebra ${rm U}(mathfrak{g}^{}_{ell})$. A similar matrix presentation for the affine Kac--Moody algebra $widehat{mathfrak{g}}^{}_{ell}$ is then used to prove an analogue of the Feigin--Frenkel theorem describing the center of the corresponding affine vertex algebra at the critical level. The proof relies on an explicit construction of a complete set of Segal--Sugawara vectors for the Lie algebra $mathfrak{g}^{}_{ell}$.
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