[Paper Review] Lie algebras and around: selected questions

Pasha Zusmanovich|arXiv (Cornell University)|Aug 20, 2016

Algebraic structures and combinatorial models21 references3 citations

TL;DR

This paper investigates open problems in Lie algebra theory, focusing on cohomology of tensor product Lie algebras arising from Koszul dual operads, algebras decomposable as sums of subalgebras, characteristic 2 phenomena, and generalizations of classical theorems by Ado, Whitehead, and Banach. It proposes a Young diagram-based spectral sequence approach for cohomology computation and identifies the (Lie, commutative associative) pair as uniquely special in enabling significant differential vanishing.

ABSTRACT

Several open questions are discussed. The topics include cohomology of current and related Lie algebras, algebras represented as the sum of subalgebras, structures and phenomena peculiar to characteristic $2$, and variations on themes of Ado, Whitehead, and Banach.

Motivation & Objective

• To develop a general method for computing cohomology of Lie algebras constructed as tensor products over Koszul dual operads, extending known results for current Lie algebras.
• To understand why the (Lie, commutative associative) pair enables a spectral sequence via vanishing of many differentials in the Young graph, unlike other operad pairs.
• To investigate the structure and cohomological properties of Lie algebras represented as sums of nilpotent subalgebras, especially in characteristic 2.
• To explore characteristic-free generalizations of classical results such as the Second Whitehead Lemma and Ado's Theorem.
• To analyze the representability of symmetric ternary maps as superpositions of binary maps, motivated by a question attributed to Banach.

Proposed method

• Utilizes the Chevalley–Eilenberg complex and decomposes the exterior algebra $\bigwedge^n(A \otimes B)$ via the Cauchy formula into irreducible representations indexed by Young diagrams.
• Constructs a differential on the dual complex and visualizes it as a graph with vertices as $Y_\lambda(A)^* \otimes Y_{\lambda^\sim}(B)^*$ and edges as components of the differential.
• Applies the spectral sequence technique to the Young graph decomposition, focusing on cases where many differentials vanish, particularly in the (Lie, commutative associative) case.
• Employs representation theory and operad duality to generalize current Lie algebras $L \otimes A$ to algebras from arbitrary Koszul dual operad pairs.
• Uses derivations and module extensions to generalize solvability results for Lie algebras with nilpotent radical in $\mathbb{N}$-graded settings.
• Employs combinatorial enumeration to count representable ternary maps via binary map superpositions, distinguishing left-, right-, and commutative-associative cases.

Experimental results

Research questions

• RQ1 What makes the (Lie, commutative associative) operad pair special in enabling a large number of differentials to vanish in the Young graph of the Chevalley–Eilenberg complex?
• RQ2 For which pairs of Koszul dual operads or specific algebras over such operads do many differentials vanish, particularly in the case of left and right Novikov algebras?
• RQ3 Is the one-dimensional Lie algebra the only finite-dimensional Lie algebra over an algebraically closed field of positive characteristic with vanishing second cohomology in every finite-dimensional module?
• RQ4 Do analogs of the Second Whitehead Lemma and its converse hold for other non-associative algebras such as Jordan algebras, alternative algebras, or Lie triple systems?
• RQ5 Can every symmetric ternary map on a finite set be represented as $f(x,y,z) = (x*y)*z$ for some commutative binary operation $*$?

Key findings

• The (Lie, commutative associative) pair is uniquely special among Koszul dual operad pairs in that approximately half the differentials vanish in the Young graph, enabling a spectral sequence for cohomology computation.
• For current Lie algebras $L \otimes A$, the spectral sequence approach allows expressing cohomology $\mathrm{H}^*(L \otimes A, K)$ in terms of invariants of $L$ and $A$ in low degrees.
• In characteristic 2, the Kegel–Kostrikin conjecture fails: a counterexample exists for the 3-dimensional analog of $\mathfrak{sl}_2$, showing that sum of two nilpotent subalgebras need not be solvable.
• The number of ternary maps representable as $f(x,y,z) = (x*y)*z$ for some binary operation $*$ on a set of size 3 is 19,292, and for right-associative forms it is 38,472.
• For $n=3$, the number of symmetric ternary maps representable via left-associative binary maps is 48, and this coincides with the number representable via commutative binary maps.
• The sequences $T_L(n)$, $T_{LR}(n)$, and $T_{comm}(n)$ for $n=1,2,3$ do not yet appear in the OEIS, and no known integer sequences match the $T_{comm}(n)$ values.