[Paper Review] Constituents of graded Lie algebras of maximal class and chain lengths of thin Lie algebras

Sandro Mattarei|arXiv (Cornell University)|Nov 8, 2020

Algebraic structures and combinatorial models25 references4 citations

TL;DR

This paper provides simplified, polynomial-based proofs for fundamental structural properties of graded Lie algebras of maximal class and thin Lie algebras, focusing on constituent lengths and diamond distributions. It establishes that the second diamond in a thin Lie algebra must occur at degree $k = 3, 5, q$, or $2q-1$, where $q$ is a power of the characteristic $p$, and determines the possible values of the key invariant $h$, the length of the second constituent, in terms of $k$ and $p$. The results unify and refine earlier classifications, particularly in positive characteristic.

ABSTRACT

Thin Lie algebras are infinite-dimensional graded Lie algebras $L=\bigoplus_{i=1}^{\infty}$, with $\dim(L_1)=2$ and satisfying a covering property: for each $i$, each nonzero $z\in L_i$ satisfies $[zL_1]=L_{i+1}$. It follows that each homogeneous components $L_i$ is either one- or two-dimensional, and in the latter case is called a diamond. Hence $L_1$ is a diamond, and if there are no other diamonds then $L$ is a graded Lie algebra of maximal class. We present simpler proofs of some fundamental facts on graded Lie algebras of maximal class, and on thin Lie algebras, based on a uniform method, with emphasis on a polynomial interpretation. Among else, we determine the possible values for the most fundamental parameter of such algebras, which is the dimension of their largest metabelian quotient.

Motivation & Objective

• To simplify and unify existing proofs of structural facts about graded Lie algebras of maximal class and thin Lie algebras using a uniform polynomial method.
• To determine the possible values of the fundamental invariant $h$, the length of the second constituent, in terms of the degree $k$ of the second diamond.
• To clarify the role of the characteristic $p$ in restricting possible diamond positions, especially for $k = q$ and $k = 2q - 1$, where $q$ is a power of $p$.
• To resolve open cases in the classification of thin Lie algebras, particularly for $k = 7$ and $p \neq 2,7$, by showing such configurations lead to contradictions.

Proposed method

• A polynomial interpretation of Lie algebra relations is used to analyze the structure of thin Lie algebras, particularly focusing on the action of $L_1$ on higher components.
• The method relies on analyzing binomial coefficients modulo $p$ to determine when certain Lie products vanish or remain nonzero, especially in the context of Equation (11).
• The proof uses a recursive analysis of Lie products such as $[vxy]$, $[vxyy]$, and $[vxxyy]$, applying identities derived from the Jacobi identity and the structure of $L_1$.
• The invariant $h$ is defined via the vanishing of specific Lie products, and its bounds are derived from the failure of binomial congruences in Equation (11) for certain $j$.
• The analysis distinguishes cases based on $k$ and $p$, using known results from [CM99], [CM04], and [CMNS96] to refine bounds on $h$.
• For $k = 2q - 1$ and $p \neq 2$, it is shown that $h = q - 1$ by contradiction, ruling out $h = q$ via Lie product expansion.

Experimental results

Research questions

• RQ1 What are the possible values for the degree $k$ of the second diamond in a thin Lie algebra over a field of positive characteristic?
• RQ2 How does the value of $h$, the length of the second constituent, depend on $k$ and the characteristic $p$?
• RQ3 Can the case $k = 7$ occur for $p \neq 2, 7$ in thin Lie algebras, and if so, under what conditions?
• RQ4 What constraints do binomial coefficient congruences modulo $p$ impose on the structure of Lie products in thin Lie algebras?
• RQ5 How do the results refine the classification of thin Lie algebras, particularly in the case $k = q$ or $k = 2q - 1$?

Key findings

• The second diamond in a thin Lie algebra must occur at degree $k = 3$, $5$, $q$, or $2q - 1$, where $q$ is a power of the characteristic $p$.
• For $k = 2q - 1$ and $p \neq 2$, the constituent length $h$ is exactly $q - 1$, and $L_{3q - 2}$ is the third diamond.
• The case $k = 7$ is ruled out for $p \neq 2, 7$ due to a contradiction arising from the non-vanishing of $[vxxyy]$ and the vanishing of $[vxx]$, violating Lie algebra relations.
• For $k = q$, the constituent length $h$ satisfies $q/2 \leq h \leq q$, and both $h = q - 1$ and $h = q$ occur depending on whether the third diamond is genuine or fake.
• For $k = 5$ and $p \neq 2,3$, the constituent length $h$ is at most 3, but $h = 3$ is ruled out, so $h = 2$, and diamonds occur in all degrees congruent to $\pm 1 \mod 6$.
• For $k = 3$, the constituent length $h$ can be 1 or 2, with $h = 1$ occurring in metabelian thin Lie algebras and $h = 2$ in those with diamonds in all odd degrees.