[Paper Review] 3-Lie-Rinehart Algebras

Ruipu Bai, Xiaojuan Li+1 more|arXiv (Cornell University)|Mar 28, 2019

Advanced Topics in Algebra10 references17 citations

TL;DR

This paper introduces 3-Lie-Rinehart algebras as a new class of algebras unifying 3-Lie algebras and Rinehart's Lie-Rinehart algebras, defined as triples $(L, A, \rho)$ where $L$ is a 3-Lie algebra and $A$-module, $\rho$ is a representation satisfying $\rho(L,L) \subseteq \text{Der}(A)$. The key contribution is constructing new 3-Lie-Rinehart algebras from existing ones and proving that crossed modules and derivations in this framework generalize classical Lie-Rinehart structures.

ABSTRACT

In this paper, we define a class of 3-algebras which are called 3-Lie-Rinehart algebras. A 3-Lie-Rinehart algebra is a triple $(L, A, ρ)$, where $A$ is a commutative associative algebra, $L$ is an $A$-module, $(A, ρ)$ is a 3-Lie algebra $L$-module and $ρ(L, L)\subseteq Der(A)$. We discuss the basic structures, actions and crossed modules of 3-Lie-Rinehart algebras and construct 3-Lie-Rinehart algebras from given algebras, we also study the derivations from 3-Lie-Rinehart algebras to 3-Lie $A$-algebras. From the study, we see that there is much difference between 3-Lie algebras and 3-Lie-Rinehart algebras.

Motivation & Objective

• To define and formalize a new class of algebras, 3-Lie-Rinehart algebras, that unify 3-Lie algebras and Rinehart's Lie-Rinehart algebras.
• To investigate the structure of actions, modules, and crossed modules within 3-Lie-Rinehart algebras.
• To construct new 3-Lie-Rinehart algebras from existing ones using tensor products and exterior algebras.
• To study derivations from 3-Lie-Rinehart algebras to 3-Lie $A$-algebras and their representation-theoretic implications.
• To establish conditions under which 3-Lie-Rinehart algebras give rise to crossed modules of Lie-Rinehart algebras.

Proposed method

• Define a 3-Lie-Rinehart algebra as a triple $(L, A, \rho)$, where $L$ is a 3-Lie algebra, $A$ is a commutative associative algebra, $L$ is an $A$-module, and $\rho: L \times L \to \text{Der}(A)$ satisfies a compatibility condition.
• Construct new 3-Lie-Rinehart algebras via $L \otimes A$ and $E$ using the given algebraic structure and representation $\rho$.
• Define the semidirect product $(L \ltimes R, A, \rho_4)$ to show that $(R, \beta)$ is an $(L,A,\rho)$-module if and only if the semidirect product is a 3-Lie-Rinehart algebra.
• Introduce the Lie-Rinehart algebra structure on $L \wedge L$ and $W(L,R,A)$, and define the associated representations $\rho_2$ and $\rho_3$.
• Define the crossed module $(L\wedge L, A, \beta, \partial)$ using the map $\partial(x \wedge y) = \overline{(\text{Ad}(x\wedge y), 0)}$ and $\beta(\overline{(\varphi, x\wedge y)}) = \varphi$.
• Prove that $\partial$ is a Lie algebra homomorphism and that $\beta$ defines an action satisfying the crossed module axioms, with $\ker(\partial)$ being the center of $L\wedge L$.

Experimental results

Research questions

• RQ1 How can 3-Lie algebras be extended to include module structures over commutative associative algebras while preserving the 3-Lie bracket axioms?
• RQ2 Under what conditions does a 3-Lie-Rinehart algebra give rise to a crossed module of Lie-Rinehart algebras?
• RQ3 What is the role of the representation $\rho$ in determining derivations and actions on 3-Lie $A$-algebras?
• RQ4 How do constructions like $L \otimes A$, $L \wedge L$, and $W(L,R,A)$ yield new 3-Lie-Rinehart algebras from a given one?
• RQ5 When does the kernel of the map $\partial$ in the crossed module structure correspond to the center of $L \wedge L$?

Key findings

• The semidirect product $(L \ltimes R, A, \rho_4)$ is a 3-Lie-Rinehart algebra if and only if $(R, \beta)$ is an $(L,A,\rho)$-module.
• Any 3-Lie $A$-algebra homomorphism from $L$ to $R$ induces a well-defined action $\beta$ of $(L,A,\rho)$ on $(R,A)$.
• The construction $L \otimes A$ with $\rho_1$ yields a 3-Lie-Rinehart algebra, as shown in Theorem 2.6.
• The exterior algebra $L \wedge L$ with $\rho_2$ forms a Lie-Rinehart algebra, as established in Theorem 2.8.
• The map $\partial: L \wedge L \to \overline{W(L,L,A)}$ defined by $\partial(x \wedge y) = \overline{(\text{Ad}(x\wedge y), 0)}$ is a Lie algebra homomorphism.
• The kernel of $\partial$ is the center of the Lie algebra $L \wedge L$, and it is closed under the $A$-module action.