Central extension of low dimensional associative and Leibniz algebras - Ab Rahman, Nurnazhifa (2020)
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.