Casimir elements and Sugawara operators for Takiff algebras

Abstract

For every simple Lie algebra g we consider the associated Takiff algebra g ℓ defined as the truncated polynomial current Lie algebra with coefficients in g. We use a matrix presentation of g ℓ to give a uniform construction of algebraically independent generators of the center of the universal enveloping algebra U(g ℓ ). A similar matrix presentation for the affine Kac-Moody algebra g ℓ 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 g ℓ .