Casimir elements and Sugawara operators for Takiff algebras

For every simple Lie algebra g, we consider the associated Takiff algebra gℓ defined as the truncated polynomial current Lie algebra with coefficients in g. We use a matrix presentation of gℓ to give a uniform construction of algebraically independent generators of the center of the universal enveloping algebra U(gℓ). A similar matrix presentation for the affine Kac–Moody algebra ĝℓ is then used to prove an analog 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 gℓ.