The center and invariants of standard filiform Lie algebras

TL;DR

This paper investigates the structure of centers and invariant rings of standard filiform Lie algebras over various fields, providing explicit generators, combinatorial results, and descriptions of their quotient fields.

Contribution

It offers explicit generators and combinatorial insights into the centers and invariants of standard filiform Lie algebras, extending results to different characteristics.

Findings

Explicit generators for invariant rings provided

Descriptions of quotient fields established

Hilbert series and combinatorial properties analyzed

Abstract

This paper describes the centers of the universal enveloping algebras and the invariant rings of the standard filiform Lie algebras over fields of characteristic zero and also over large enough prime characteristic. We determine explicit generators for the quotient fields and also a compact form for the generators for the invariants rings. We prove several combinatorial results concerning the Hilbert series of these algebras.