Central extension of low dimensional associative and Leibniz algebras
Leibniz algebras was introduced by Louis Loday due to several considerations in algebraic K-theory. The cyclic cohomology was somehow related to Lie algebra homology. Chevalley-Eilenberg chain complex that included in Lie algebras which the exterior powers of the Lie algebra was involved. There w...
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| Main Author: | |
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| Format: | Thesis |
| Language: | English |
| Published: |
2020
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| Subjects: | |
| Online Access: | http://psasir.upm.edu.my/id/eprint/104715/1/NURNAZHIFA%20BINTI%20AB%20RAHMAN%20-%20IR.pdf |
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| Summary: | Leibniz algebras was introduced by Louis Loday due to several considerations in
algebraic K-theory. The cyclic cohomology was somehow related to Lie algebra
homology. Chevalley-Eilenberg chain complex that included in Lie algebras which
the exterior powers of the Lie algebra was involved. There was a new complex
which a noncommutative generalization was categorized in Lie algebra. It was
called Leibniz homology as Loday is the founder. As the Lie algebra homology was
related to the cyclic homology, the Leibniz homology somehow was related to the
Hochschild homology.
The main purpose of this thesis is to apply the Skjelbred-Sund method and find a list
of isomorphism classes for associative and Leibniz algebras. The method used is to
find the high dimension of algebras by using the list of low dimensional algebras. It
deals with one dimensional central extension of low dimensional algebras. To get
the central extension of algebras, the condition q?\C(L) = 0 need to be satisfied. If
the condition is not satisfied, there is no central extension of one dimensional algebra
which means this method is inapplicable. In addition, some extension invariants
such as center, radical, coboundary, centroid, maximum commutative subalgebra,
maximum abelian subalgebra and second cohomology are needed to investigate the
isomorphism problem. As the results, seven isomorphism classes of three dimensional
associative algebras, eight isomorphism classes of three dimensional Leibniz
algebras and 13 isomorphism classes of four dimensional Leibniz algebra are provided. |
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