Classification and Construction of Low-dimensional Hom-Lie Algebras and Ternary Hom-Nambu-Lie Algebras

This thesis concerns the construction and classification of low-dimensional Hom-Lie algebras and ternary Hom-Nambu-Lie algebras. A classification of 3-dimensional Hom-Lie algebras is given for nilpotent linear endomorphism, as a twisting map, and a construction of 4-dimensional Hom-Lie algebras is done. Results on the dimension of the space of endomorphisms that turn a skew-symmetric algebra into a Hom-Lie algebra are also given in this thesis. A class of 3-dimensional ternary Hom-Nambu-Lie algebras with nilpotent linear maps are constructed and classified. In Chapter 2, we derive conditions for an arbitrary n-dimensional algebra to be a Hom-Lie algebra, in the form of a system of polynomial equations, containing both structure constants of the skew-symmetric bilinear map and constants describing the twisting linear endomorphism. When the algebra is 3 or 4-dimensional, we describe the realisation of Hom-Lie algebras when the dimension of the space of such linear endomorphisms, as vector spaces, is minimum. For the 3-dimensional case we give all possible families of 3-dimensional Hom-Lie algebras arising from a general nilpotent linear endomorphism constructed up to isomorphism together with non-isomorphic canonical representatives for all the families in that case. We further give a list of 4-dimensional Hom-Lie algebras arising from general nilpotent linear endomorphisms. In Chapter 3, we describe the dimension of the space of possible linear endomorphisms that turn skew-symmetric three-dimensional algebras into Hom-Lie algebras. We find a correspondence between the rank of a matrix containing the structure constants of the bilinear product and the dimension of the space of Hom-Lie structures. Examples from classical complex Lie algebras are given to demonstrate this correspondence. In Chapter 4, the space of possible Hom-Lie structures on complex 4-dimensional Lie algebras is considered in terms of linear maps that turn the Lie algebras into Hom-Lie algebras. Hom-Lie structures and automorphism groups on the representatives of isomorphism classes of complex 4-dimensional Lie algebras are described. In Chapter 5, we construct ternary Hom-Nambu-Lie algebras from Hom-Lie algebras through a process known as induction. The induced algebras are constructed from a class of Hom-Lie algebra with nilpotent linear map. The families of ternary Hom-Nambu-Lie arising in this way of construction are classified for a given class of nilpotent linear maps. In addition, some results giving conditions on when morphisms of Hom-Lie algebras can still remain morphisms for the induced ternary Hom-Nambu-Lie algebras are given.