Classical $\mathcal{W}$-algebras for centralizers

Abstract

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 $\mathfrak{a}$.

15 pages; minor revision

References in corpus (4)

• Classical W-algebras for gl_N and associated integrable Hamiltonian hierarchies
• Classical affine W-algebras and the associated integrable Hamiltonian hierarchies for classical Lie algebras
• Elementary invariants for centralizers of nilpotent matrices
• Classical W-algebras in types A, B, C, D and G

Cited by in corpus (2)

• $W$-algebras associated with centralizers in type $A$
• Generalized finite and affine $W$-algebras in type $A$