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Extension of Torsors and Curved Maurer Cartan Equation


In this thesis we will study the extension problem of (nilpotent) G-torsors.

In chapter 1, we will review the Maurer Cartan equation of a DGLA and go through some

examples of Maurer Cartan equations in deformation of different algebraic structures.

In chapter 2, we will define L infinity algebras, which are generalizations of DGLAs. We will also

state the homotopy transfer of structure theorem and the formal Kuranishi theorem, which

are used throughout the whole thesis.

In chapter 3, we will briefly go through Getzler's result on the unique horn filling of Deligne-

Getzler infinite-groupoids, which gives us a generalization (on L infinity algebras) of the Baker-Campbell-

Hausdorff formula for a Lie algebra.

In chapter 4, we will state Hinich's result on descent of Deligne groupoid and go through

Fiorenza, Manetti, and Martinengo's result on the special case when we have a semicosimplicial

Lie algebra, i.e. the solutions to the group valued cocycle condition are exactly the

Maurer Cartan solutions on the L infinity total complex and equivalence of cocycles are exactly

equivalence of Maurer Cartan solutions. We will then show an example of this result on

deformation of (nilpotent) G-torsors.

In chapter 5, we will apply Fiorenza, Manetti, and Martinengo's result on the extension

problem of (nilpotent) G-torsors and show that solutions to the curved cocycle condition

that gives the G-torsors extensions are exactly the curved Maurer Cartan solutions of a

curved L infinity algebra and equivalence of extensions are exactly equivalence of curved Maurer

Cartan solutions.

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