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 $\mathrm{U}({\mathfrak{g}}_{\ell }^{})$. 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 ${\mathfrak{g}}_{\ell }^{}$.

16 pages, minor changes

References in corpus (3)

• The $R$-matrix presentation for the Yangian of a simple Lie algebra
• Elementary invariants for centralizers of nilpotent matrices
• Center at the critical level for centralizers in type $A$