Quantized symplectic actions and 𝑊-algebras

Abstract

With a nilpotent element in a semisimple Lie algebra g \mathfrak {g} one associates a finitely generated associative algebra W \mathcal {W} called a W W -algebra of finite type . This algebra is obtained from the universal enveloping algebra U ( g ) U(\mathfrak {g}) by a certain Hamiltonian reduction. We observe that W \mathcal {W} is the invariant algebra for an action of a reductive group G G with Lie algebra g \mathfrak {g} on a quantized symplectic affine variety and use this observation to study W \mathcal {W} . Our results include an alternative definition of W \mathcal {W} , a relation between the sets of prime ideals of W \mathcal {W} and of the corresponding universal enveloping algebra, the existence of a one-dimensional representation of W \mathcal {W} in the case of classical g \mathfrak {g} and the separation of elements of W \mathcal {W} by finite-dimensional representations.