[Paper Review] Non-degenerate Killing forms on Hom-Lie superalgebras

Abdoreza Armakan, M. R. Farhangdoost+1 more|arXiv (Cornell University)|Oct 5, 2020

Advanced Topics in Algebra51 references4 citations

TL;DR

This paper investigates non-degenerate Killing forms on Hom-Lie superalgebras, establishing that such forms exist if and only if the algebra is a direct product of classical simple Hom-Lie superalgebras with non-degenerate Killing forms. The authors classify invariant bilinear forms, prove that simple Hom-Lie superalgebras have no nontrivial left or right ideals, and derive Cartan-type criteria for non-degeneracy, linking it to reductivity and complete reducibility of representations.

ABSTRACT

In this paper we investigate some important basic properties of simple Hom-Lie superalgebras and show that a Hom-Lie superalgebra does not have any left or right nontrivial ideals. Moreover, we classify invariant bilinear forms on a given simple Hom-Lie superalgebra. Then we study the Killing forms on a Hom-Lie algebra which are examples of the invariant bilinear forms. Making use of the Killing forms, we find conditions for a Hom-Lie superalgebra to be classical. Furthermore, we check the conditions in which the Killing form of a Hom-Lie superalgebra is non-degenerate.

Motivation & Objective

• To investigate the structure of simple Hom-Lie superalgebras and prove they admit no nontrivial left or right ideals.
• To classify invariant bilinear forms on simple Hom-Lie superalgebras, particularly focusing on Killing forms.
• To establish necessary and sufficient conditions for the Killing form of a Hom-Lie superalgebra to be non-degenerate.
• To determine when a Hom-Lie superalgebra is classical, based on properties of its Killing form.
• To extend Cartan’s criterion to the Hom-Lie superalgebra setting using non-degeneracy of the Killing form.

Proposed method

• Use of graded ideals and orthogonality with respect to a supersymmetric, non-degenerate bilinear form to decompose the algebra into minimal simple components.
• Application of the Hom-Jacobi identity and skew-symmetry to analyze the structure of Hom-Lie superalgebras.
• Definition and analysis of the Killing form via the trace of adjoint maps in the context of Hom-algebras.
• Proof that a non-degenerate Killing form implies the absence of non-zero commutative graded ideals.
• Use of induction on dimension to show that a Hom-Lie superalgebra with non-degenerate form decomposes into orthogonal, simple, and classical components.
• Application of representation theory to show that complete reducibility of the $γ_{\bar{1}}$-representation over $\mathfrak{g}_{\bar{0}}$ is necessary for non-degeneracy.

Experimental results

Research questions

• RQ1 Under what conditions is the Killing form of a Hom-Lie superalgebra non-degenerate?
• RQ2 When is a Hom-Lie superalgebra classified as classical, based on its Killing form?
• RQ3 Does a Hom-Lie superalgebra admit any nontrivial left or right ideals, particularly in the simple case?
• RQ4 How do invariant bilinear forms, especially the Killing form, behave under decomposition into simple components?
• RQ5 What structural properties (e.g., reductivity, complete reducibility) are implied by the non-degeneracy of the Killing form?

Key findings

• A Hom-Lie superalgebra has no nontrivial left or right ideals, and any such ideal is necessarily graded.
• The Killing form of a Hom-Lie superalgebra is non-degenerate if and only if the algebra is isomorphic to a direct product of classical simple Hom-Lie superalgebras with non-degenerate Killing forms.
• The restriction of the Killing form to each simple component is non-degenerate, implying each component is classical.
• The even part $\mathfrak{g}_{\bar{0}}$ is reductive and the representation of $\mathfrak{g}_{\bar{0}}$ on $\mathfrak{g}_{\bar{1}}$ is completely reducible if the Killing form is non-degenerate.
• The bilinear form is supersymmetric, and the algebra decomposes as an orthogonal direct sum of minimal simple graded ideals with respect to the form.