#### Title

Calculating the Cohomology of a Lie Algebra Using Maple and the Serre Hochschild Spectral Sequence

#### Date of Award

12-2018

#### Degree Type

Report

#### Degree Name

Master of Science (MS)

#### Department

Mathematics and Statistics

#### First Advisor

Ian Anderson

#### Second Advisor

Mark Fels

#### Third Advisor

John R. Stevens

#### Abstract

Lie algebra cohomology is an important tool in many branches of mathematics. It is used in the Topology of homogeneous spaces, Deformation theory, and Extension theory. There exists extensive theory for calculating the cohomology of semi simple Lie algebras, but more tools are needed for calculating the cohomology of general Lie algebras. To calculate the cohomology of general Lie algebras, I used the symbolic software program called Maple. I wrote software to calculate the cohomology in several different ways. I wrote several programs to calculate the cohomology directly. This proved to be computationally expensive as the number of differential forms used in each step of the direct calculation is equal to a binomial expansion. The second method used was the Serre--Hochschild spectral sequence. The Serre--Hochschild spectral sequence breaks the problem up into a grid of smaller problems that converge to the total cohomology of the Lie algebra.

#### Recommended Citation

Kullberg, Jacob, "Calculating the Cohomology of a Lie Algebra Using Maple and the Serre Hochschild Spectral Sequence" (2018). *All Graduate Plan B and other Reports*. 1321.

https://digitalcommons.usu.edu/gradreports/1321

#### Included in

Algebraic Geometry Commons, Numerical Analysis and Computation Commons, Numerical Analysis and Scientific Computing Commons

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