Casimir elements and center at the critical level for Takiff algebras

A. I. Molev

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 $\mathrm{U}({\mathfrak{g}}_{\ell }^{})$. A similar matrix presentation for the affine Kac–Moody algebra ${\hat{\mathfrak{g}}}_{\ell }^{}$ 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 ${\mathfrak{g}}_{\ell }^{}$.