[Paper Review] Cohomology of Hom-Leibniz and $n$-ary Hom-Nambu-Lie superalgebras

El Kadri Abdaoui, Sami Mabrouk+1 more|arXiv (Cornell University)|Jun 14, 2014

Advanced Topics in Algebra25 references6 citations

TL;DR

This paper develops cohomology theory for Hom-Leibniz and $n$-ary Hom-Nambu-Lie superalgebras, generalizing Takhtajan's approach to twisted and graded settings. It constructs the $q$-deformed Heisenberg-Virasoro superalgebra of Hom-type and computes its derivations and second cohomology group, establishing a cohomological link between $n$-ary Hom-Nambu-Lie and Hom-Leibniz superalgebras via a derived complex structure.

ABSTRACT

The aim of this paper is to study the cohomology of Hom-Leibniz superalgebras. We construct the $q$-deformed Heisenberg-Virasoro superalgebra of Hom-type and provide as application the computations of the derivations and second cohomology group. Moreover, we extend to graded case the Takhtajan's construction of a cohomology of $n$-ary Hom-Nambu-Lie algebras starting from cohomology of Hom-Leibniz algebras.

Motivation & Objective

• To develop a cohomology theory for Hom-Leibniz superalgebras in the graded setting.
• To extend Takhtajan’s construction of $n$-ary Hom-Nambu-Lie cohomology from Hom-Leibniz superalgebra cohomology.
• To compute the derivations and second cohomology group of the $q$-deformed Heisenberg-Virasoro superalgebra of Hom-type.
• To establish a structural relationship between $n$-ary Hom-Nambu-Lie superalgebras and Hom-Leibniz superalgebras via cohomological methods.
• To generalize the $n$-ary Nambu identity and its cohomological framework to the Hom-twisted and graded superalgebraic setting.

Proposed method

• Define Hom-Leibniz superalgebras via a $eta$-twisted super-Jacobi identity involving an even endomorphism $\alpha$.
• Introduce representations and derivations of Hom-Leibniz superalgebras using the adjoint map $ad_x(y) = [x,y]$.
• Construct the $q$-deformed Heisenberg-Virasoro superalgebra as a specific example of a Hom-Leibniz superalgebra.
• Define a cohomology complex for Hom-Leibniz superalgebras using a differential $\delta$ acting on multilinear cochains.
• Generalize the $n$-ary Nambu identity to the Hom-twisted case and define a cohomology for $n$-ary Hom-Nambu-Lie superalgebras.
• Establish a cochain map $\Delta$ from Hom-Leibniz superalgebra cochains to $n$-ary Hom-Nambu-Lie superalgebra cochains, proving $d \circ \Delta = \Delta \circ \delta$.

Experimental results

Research questions

• RQ1 How can cohomology be defined for Hom-Leibniz superalgebras in the graded setting?
• RQ2 What is the structure of the second cohomology group and derivations of the $q$-deformed Heisenberg-Virasoro superalgebra of Hom-type?
• RQ3 How does the cohomology of $n$-ary Hom-Nambu-Lie superalgebras relate to that of Hom-Leibniz superalgebras?
• RQ4 Can Takhtajan’s construction of $n$-ary cohomology be generalized to the Hom-twisted and graded case?
• RQ5 What is the role of the cochain map $\Delta$ in relating the cohomologies of $n$-ary Hom-Nambu-Lie and Hom-Leibniz superalgebras?

Key findings

• The $q$-deformed Heisenberg-Virasoro superalgebra of Hom-type is constructed as a specific example of a Hom-Leibniz superalgebra.
• The derivations of the $q$-deformed Heisenberg-Virasoro superalgebra are computed explicitly via the cohomological framework.
• The second cohomology group of the $q$-deformed Heisenberg-Virasoro superalgebra is computed, providing information on central extensions.
• A cohomology complex for $n$-ary Hom-Nambu-Lie superalgebras is defined using a differential $\delta$ satisfying $\delta^2 = 0$.
• A cochain map $\Delta$ is constructed such that $d \circ \Delta = \Delta \circ \delta$, linking the cohomologies of $n$-ary Hom-Nambu-Lie and Hom-Leibniz superalgebras.
• The cohomological framework generalizes Takhtajan’s construction to the graded and Hom-twisted setting, preserving the $n$-ary Nambu identity structure.