Casimir elements and Sugawara operators for Takiff algebras

Abstract: 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 $\mathfrak{g}^{}_{\ell}$.