[Paper Review] A Class of Strongly Homotopy Lie Algebras with Simplified sh-Lie Structures

Samer Al-Ashhab|ArXiv.org|Aug 17, 2003

Advanced Topics in Algebra4 references4 citations

TL;DR

This paper constructs a class of strongly homotopy Lie (sh-Lie) algebras where higher-order structure maps ($l_n$ for $n > 3$) vanish, and $l_2$ and $l_3$ are nontrivial only in degree 0. By leveraging a chain homotopy equivalence and a boundary condition on $l_2 l_1$, the authors show that the sh-Lie structure simplifies significantly: only $l_2$ and $l_3$ are non-zero in degree 0, and all higher maps vanish, even when $l_2(c,b)$ is not zero—extending prior results under weaker assumptions.

ABSTRACT

It is known that a single mapping defined on one term of a differential graded vector space extends to a strongly homotopy Lie algebra structure on the graded space when that mapping satisfies two conditions. This strongly homotopy Lie algebra is nontrivial (it is not a Lie algebra); however we show that one can obtain an sh-Lie algebra where the only nonzero mappings defining it are the lower order mappings. This structure applies to a significant class of examples. Moreover in this case the graded space can be replaced by another graded space, with only three nonzero terms, on which the same sh-Lie structure exists.

Motivation & Objective

• To construct a class of sh-Lie algebras with simplified higher-order structure maps.
• To show that $l_n \equiv 0$ for $n > 3$, and $l_2 = l_3 = 0$ in degrees $> 0$, under minimal assumptions.
• To generalize prior results by relaxing the condition $l_2(c,b) = 0$ to only requiring $l_2(c,b)$ to be a boundary.
• To demonstrate that such sh-Lie structures exist on a minimal graded space with only three non-zero terms.
• To provide explicit constructions in geometric contexts, including symplectic manifolds and cohomology complexes.

Proposed method

• Define $l_2$ inductively via $l_2 = -s \circ l_2 l_1$, using a chain homotopy $s$ satisfying $\lambda \circ \eta - \text{id} = l_1 \circ s$ in degree 0 and $-\text{id} = l_1 \circ s + s \circ l_1$ in higher degrees.
• Use Lemma 2.1 to confirm that $l_2 l_1$ is a boundary, ensuring $l_2$ is well-defined as a chain map.
• Prove $l_2 = 0$ on $X_1 \otimes X_1$ and $X_2 \otimes X_0$ by showing $l_2 l_1$ vanishes on these components.
• Show $l_3 = 0$ on $X_1 \otimes X_0 \otimes X_0$ and higher degrees by verifying $l_2 l_2 + l_3 l_1$ vanishes via Jacobi identity and Lie derivative identities.
• Construct explicit examples using Poisson brackets on symplectic manifolds and cohomology complexes.
• Demonstrate that the sh-Lie structure can be transferred to a minimal model with only three non-zero terms, preserving the same $l_2$ and $l_3$ maps in degree 0.

Experimental results

Research questions

• RQ1 Can an sh-Lie algebra be constructed such that all higher structure maps $l_n$ for $n > 3$ vanish, even when $l_2(c,b) \neq 0$?
• RQ2 Under what conditions can $l_2$ and $l_3$ be non-zero only in degree 0, with all higher-degree maps vanishing?
• RQ3 Is it possible to reduce a given sh-Lie algebra to a minimal model with only three non-zero graded components while preserving the essential structure?
• RQ4 How does the chain homotopy $s$ and the boundary condition $l_2 l_1$ being a boundary enable the simplification of the sh-Lie structure?
• RQ5 Can the construction be applied to geometric examples such as symplectic manifolds with Poisson brackets?

Key findings

• The paper constructs an sh-Lie algebra where $l_n \equiv 0$ for $n > 3$, and $l_2 = l_3 = 0$ in all degrees $> 0$, simplifying the structure significantly.
• The $l_2$ map is non-zero only in degree 0, defined as $l_2(P\nu, Q\nu) = \omega(\mathbf{E}(Q), \mathbf{E}(P))\nu$, and vanishes on $X_1 \otimes X_1$, $X_2 \otimes X_0$, and their symmetric counterparts.
• The $l_3$ map is non-zero only in degree 0, with $l_3(P\nu, Q\nu, R\nu)$ being non-vanishing, while $l_3 = 0$ in all higher degrees.
• The sh-Lie structure can be transferred to a minimal model with only three non-zero terms: $X_0$, $X_1$, and $X_2$, preserving the same $l_2$ and $l_3$ maps in degree 0.
• In the symplectic manifold example, $l_2(f, \beta) = L_{\beta^\#}f$ in degree 1, and $l_2 = 0$ in higher degrees, with $l_3 = 0$ on $X_1 \otimes X_0 \otimes X_0$ due to the Jacobi identity and Lie derivative identities.
• The construction holds under the weaker condition that $l_2(c,b)$ is a boundary rather than zero, generalizing prior results that required $l_2(c,b) = 0$.