Deformations of Ideals in Lie Algebroids

by Ilias Ermeidis
Doctoral thesis
Date of Examination:2025-02-10
Date of issue:2025-07-31
Advisor:Prof. Dr. Madeleine Jotz
Referee:Prof. Dr. Madeleine Jotz
Referee:Prof. Dr. Chenchang Zhu
Persistent Address: http://dx.doi.org/10.53846/goediss-11406

 

 

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Abstract

English

In this dissertation, we address the deformation problem of ideals in Lie algebras and its generalization(s) in the context of Lie algebroids. Lie ideals lie at the heart of the representation theory of Lie algebras, as well as their classification theory. The upshot of deformation theory, as the infinitesimal study of moduli spaces, is to better understand the local geometry of the space of algebro-geometric objects of a specific kind. We attach to every Lie ideal a differential graded Lie algebra such that its Maurer-Cartan elements are in one-to-one correspondence with the (small) deformations of the ideal. Furthermore, we investigate how this theory is related to other well-known deformation theories, and, in addition, we attach to each Lie ideal a Lie-infinity algebra that controls the deformations of both the ideal and the ambient Lie bracket simultaneously. Under appropriate assumptions regarding the low levels of the deformation cohomology of a Lie ideal, we obtain (topological) rigidity and stability results. By translating the main technique used to solve the deformation problem of a Lie ideal inside a Lie algebra into differential graded geometric terms, we pave the way towards a generalization of this theory to bundles of Lie ideals within Lie algebroids. Finally, given that the most general (and appropriate) notion of an ideal in a Lie algebroid is equivalent to that of a double Lie subalgebroid of its tangent double bundle, we discuss the corresponding deformation problem, explain the necessary ingredients to tackle it, and describe the progress that has been made thus far.
Keywords: ideals; Lie algebras; deformation theory; Lie-infinity algebras; rigidity; stability; moduli space; Lie algebroids; differential graded geometry; higher Lie theory; double structures
 
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