It is shown that every abelian regular Lie group is a quotient of its Lie algebra via the exponential mapping. This paper is a sequel of [3], see also [4], chapter VIII, where a regular Lie group is defined as a smooth Lie group modeled on convenient vector spaces such that the right logarithmic derivative has a smooth inverse Evol : C(R, g) → C(R, G), the canonical evolution operator, where g ...

We present an explicit description of the 'fine group gradings' (i.e. group gradings which cannot be further refined) of the real forms of the semisimple Lie algebras sl(4, C), sp(4, C), and o(4, C). All together 12 real Lie algebras are considered, and the total of 44 of their fine group gradings are listed. The inclusions sl(4, C) ⊃ sp(4, C) ⊃ o(4, C) are an important tool in our presentation...

We present an explicit description of the 'fine gradings' (i.e. grad-ings which cannot be further refined) of the real forms of the semisimple Lie algebras sl(4, C), sp(4, C), and o(4, C). All together 12 real Lie algebras are considered, and the total of 44 of their fine gradings are listed. The inclusions sl(4, C) ⊃ sp(4, C) ⊃ o(4, C) are an important tool in our presentation. Systematic use ...

Let B be a representation-finite C-algebra. The Z-Lie algebra L(B) associated with B has been defined by Ch. Riedtmann in [17]. If B is representation-directed there is another Z-Lie algebra associated with B defined by C. M. Ringel in [20] and denoted by K(B). We prove that the Lie algebras L(B) and K(B) are isomorphic for any representation-directed C-algebra B. 2000 Mathematics Subject Class...

The notion of strongly Lie zero-product preserving maps on normed algebras as a generalization of Lie zero-product preserving maps are dened. We give a necessary and sufficient condition from which a linear map between normed algebras to be strongly Lie zero-product preserving. Also some hereditary properties of strongly Lie zero-product preserving maps are presented. Finally the second dual of...

We use a method of commutative algebra to describe the affine variety $\textrm{HLie}_{m}(\mathfrak{gl}_{n}(\mathbb{C}))$ all multiplicative Hom-Lie algebras on general linear Lie $\mathfrak{gl}_{n}(\mathbb{C})$, showing that $\textrm{HLie}_{m}(\mathfrak{gl}_{2}(\mathbb{C}))$ consists two 1-dimensional and one 3-dimensional irreducible components. also prove $\textrm{HLie}_{m}(\mathfrak{gl}_{n}(...

Models of all the gradings on the exceptional simple Lie algebras induced by Jordan subgroups of their groups of automorphisms are provided.

We establish some results about large restricted Lie algebras similar to those known in the Group Theory. As an application, we use this group-theoretic approach to produce some examples of restricted as well as ordinary Lie algebras which can serve as counterexamples for various Burnside type questions.

This paper studies the relationship between representations of a Lie group and representations of its Lie algebra. We will make the correspondence in two steps: First we shall prove that a given representation of a Lie group will provide us with a corresponding representation of its Lie algebra. Second, we shall go backwards and see whether a given representation of a Lie algebra will have a co...