Computational Construction of Weierstrass Sections for Semi-Invariant Polynomial Functions on Lie Algebras

Abstract

In this paper, we present a computational approach to construct Weierstrass sections for semi-invariant polynomial functions on Lie algebras, extending the foundational work of Bourbaki and Popov. We focus on simple Lie algebras of type B, C, or D, and their associated parabolic subalgebras, particularly those with Levi factors composed of successive blocks of size two. Our method extends the notion of Weierstrass sections introduced by Popov, enabling us to explicitly construct these sections and establish their polynomiality. Furthermore, we demonstrate how these sections facilitate the linearization of semi-invariant generators. Central to our approach is the construction of an adapted pair, akin to a principal sl_2-triple in the non-reductive case. We provide computational algorithms and implementations for constructing these Weierstrass sections, offering a novel avenue for research in Lie algebra theory and algebraic geometry.