[Paper Review] Representations and derivations of Hom-Lie conformal superalgebras

Shuangjian Guo, Lihong Dong+1 more|arXiv (Cornell University)|Jul 8, 2018

Advanced Topics in Algebra7 references3 citations

TL;DR

This paper develops representation theory, cohomology, and derivation theory for Hom-Lie conformal superalgebras, establishing foundational tools for studying deformations and structural properties. It proves that generalized derivations decompose into quasiderivations and quasicentralizers, and shows that under surjectivity of the twisting map and trivial center, the quasicentralizer component vanishes, simplifying the derivation algebra structure.

ABSTRACT

In this paper, we introduce a representation theory of Hom-Lie conformal superalgebras and discuss the cases of adjoint representations. Furthermore, we develop cohomology theory of Hom-Lie conformal superalgebras and discuss some applications to the study of deformations of regular Hom-Lie conformal superalgebras. Finally, we introduce derivations of multiplicative Hom-Lie conformal superalgebras and study their properties.

Motivation & Objective

• To establish a representation theory for Hom-Lie conformal superalgebras, including adjoint representations.
• To develop cohomology theory for Hom-Lie conformal superalgebras and apply it to deformation theory of regular algebras.
• To introduce and analyze derivations in multiplicative Hom-Lie conformal superalgebras, particularly their algebraic structure and closure properties.
• To define generalized derivations and investigate their decomposition into quasiderivations and quasicentralizers.
• To determine conditions under which the quasicentralizer component vanishes, especially when the center is trivial.

Proposed method

• Introduces representations of Hom-Lie conformal superalgebras via Hom-conformal linear maps satisfying twisted Jacobi-type identities.
• Develops cohomology theory using cochain complexes with coefficients in representations, enabling deformation analysis.
• Defines derivations as Hom-conformal linear maps satisfying a twisted Leibniz rule involving the twisting map α.
• Introduces generalized derivations as sums of quasiderivations and quasicentralizers, with component-wise conditions on λ-brackets.
• Uses surjectivity of α and trivial center to derive structural constraints on the derived algebra of derivations.
• Applies the generalized derivation decomposition to prove that [C(R)λQC(R)] ⊆ Chom(R,Z(R))[λ], leading to vanishing when Z(R)=0.

Experimental results

Research questions

• RQ1 How can a representation theory be formulated for Hom-Lie conformal superalgebras, and what are the properties of the adjoint representation?
• RQ2 What is the structure of the cohomology complex for Hom-Lie conformal superalgebras, and how does it relate to deformation theory?
• RQ3 Under what conditions does the space of generalized derivations decompose into quasiderivations and quasicentralizers?
• RQ4 What is the algebraic structure of the space of derivations in multiplicative Hom-Lie conformal superalgebras?
• RQ5 When does the quasicentralizer component of generalized derivations vanish, particularly in the case of trivial center?

Key findings

• Generalized derivations of a multiplicative Hom-Lie conformal superalgebra decompose into quasiderivations and quasicentralizers: GDer(R) = QDer(R) + QC(R).
• The space of quasicentralizers QC(R) is closed under the λ-bracket when the twisting map α is surjective and the center Z(R) is trivial.
• For any multiplicative Hom-Lie conformal superalgebra with surjective α, the bracket [C(R)λQC(R)] is contained in Chom(R,Z(R))[λ], the space of Hom-conformal maps into the center.
• If the center Z(R) is trivial, then [C(R)λQC(R)] = 0, implying that the quasicentralizer component does not contribute to the Lie conformal structure.
• When Z(R) = 0, QC(R) forms a Hom-Lie conformal superalgebra if and only if [QC(R)λQC(R)] = 0, providing a structural criterion for closure.
• The direct sum of the space of derivations is itself a Hom-Lie conformal superalgebra, and any derivation gives rise to a derivation extension of the algebra.