The center of the universal enveloping algebras of small-dimensional nilpotent Lie algebras in prime characteristic

TL;DR

This paper characterizes the centers of universal enveloping algebras for small-dimensional nilpotent Lie algebras over fields of prime characteristic, confirming Braun's conjecture for this class.

Contribution

It provides explicit descriptions of the centers for nilpotent Lie algebras up to dimension six, including exceptional cases, and proves their isomorphism to the Poisson center.

Findings

Centers are the integral closure of the $p$-center-generated algebra in most cases.

In three exceptional cases, the centers have additional generators.

The centers are isomorphic to the Poisson centers, confirming Braun's conjecture.

Abstract

We describe the centers of the universal enveloping algebras of nilpotent Lie algebras of dimension at most six over fields of prime characteristic. If the characteristic is not smaller than the nilpontency class, then the center is the integral closure of the algebra generated over the $p$-center by the same generators that also occur in characteristic zero. Except for three examples (two of which are standard filiform), this algebra is already integrally closed and hence it coincides with the center. In the case of these three exceptional algebras, the center has further generators. Then we show that the center of the universal enveloping algebra of the algebras investigated in this paper is isomorphic to the Poisson center (the algebra of invariants under the adjoint representation). This shows that Braun's conjecture is valid for this class of Lie algebras.