[Paper Review] On wild Lie algebras

Ievgen Makedonskyi|arXiv (Cornell University)|Feb 7, 2012

Algebraic structures and combinatorial models6 references3 citations

TL;DR

This paper classifies finite-dimensional Lie algebras over an algebraically closed field of characteristic zero as either tame or controlled wild based on the complexity of their representation classification problem. It proves that only five classes of Lie algebras are tame: semisimple algebras, the one-dimensional algebra, their direct sums, $\mathfrak{sl}_2 \rightthreetimes I$ with the two-dimensional irreducible module $I$, and direct sums involving these. All other Lie algebras are controlled wild, meaning their representation problem is as complex as classifying pairs of matrices up to simultaneous similarity.

ABSTRACT

We give a criterion of tameness and wildness for a finite-dimensional Lie algebra over an algebraically closed field.

Motivation & Objective

• To determine which finite-dimensional Lie algebras over an algebraically closed field of characteristic zero have tame or wild representation classification problems.
• To establish a complete classification of tame Lie algebras, distinguishing them from controlled wild ones.
• To analyze the structure of Lie algebras with abelian or non-abelian radicals via quiver representations and cohomological methods.
• To prove that any Lie algebra with a non-abelian radical is wild, extending the classification to all finite-dimensional Lie algebras.

Proposed method

• Uses the Levi decomposition $\widehat{L} = L \rightthreetimes R$, where $L$ is semisimple and $R$ is the radical, to analyze the structure of Lie algebras.
• Applies Schur's lemma and complete reducibility of semisimple Lie algebra representations to decompose modules into tensor products of irreducible components.
• Reduces the representation problem of $\widehat{L}$ to quiver representations with relations, particularly analyzing $\operatorname{Hom}(I \otimes M, M)$ for abelian ideals $I$.
• Employs cohomological techniques, including the vanishing of second cohomology groups, to show that certain extensions split and thus preserve structural properties.
• Analyzes the action of the one-dimensional quotient $R/[R,R]$ on irreducible modules to determine wildness via quiver models.
• Uses known results on wildness of quiver problems (e.g., $\alpha_2\beta = \beta\alpha_1$) to prove wildness of certain Lie algebras.

Experimental results

Research questions

• RQ1 Which finite-dimensional Lie algebras over an algebraically closed field of characteristic zero have a tame classification problem for their finite-dimensional representations?
• RQ2 What structural conditions on a Lie algebra ensure that its representation problem is wild or controlled wild?
• RQ3 How does the action of the radical $R$ and its quotient $R/[R,R]$ affect the tameness or wildness of the representation problem?
• RQ4 Can the representation theory of Lie algebras with non-abelian radicals be reduced to quiver problems, and when does this lead to wildness?
• RQ5 Why is $\mathfrak{sl}_2 \rightthreetimes I$ with the two-dimensional irreducible module $I$ the only non-semisimple tame Lie algebra beyond direct sums?

Key findings

• There are exactly five classes of tame finite-dimensional Lie algebras over an algebraically closed field of characteristic zero.
• Semisimple Lie algebras are tame, as established by classical representation theory.
• The one-dimensional Lie algebra is tame, and so are all direct sums of semisimple algebras and the one-dimensional algebra.
• $\mathfrak{sl}_2 \rightthreetimes I$, where $I$ is the two-dimensional irreducible module, is tame, and so are its direct sums with semisimple algebras.
• All other finite-dimensional Lie algebras are controlled wild, including all solvable Lie algebras of dimension >1 and all Lie algebras with non-abelian radicals.
• The representation problem for $\mathfrak{sl}_2 \rightthreetimes I$ is tame because its representations decompose into tensor products of irreducible $\mathfrak{sl}_2$-modules and Jordan blocks, preserving indecomposability.