The theory of Lie algebras has many applications in mathematics and physics. One possible way of generalizing the theory of Lie algebras is to develop the theory of Lie-like algebras algebras, where the notion of a Lie-like algebras algebra was introduced in [4]. One of Lie’s Theorems claims that the only irreducible representations of a solvable Lie algebra over an algebraically closed field k...

We generalise the notion of coherent states to arbitrary Lie algebras by making an analogy with the GNS construction in C∗-algebras. The method is illustrated with examples of semisimple and non-semisimple finite dimensional Lie algebras as well as loop and Kac-Moody algebras. A deformed addition on the parameter space is also introduced simplifying some expressions and some applications to con...

Nongraded infinite-dimensional Lie algebras appeared naturally in the theory of Hamiltonian operators, the theory of vertex algebras and their multi-variable analogues. They play important roles in mathematical physics. This survey article is written based on the author’s seminar talks on nongraded infinite-dimensional simple Lie algebras. The key constructional ingredients of our Lie algebras ...

There have been a number of mathematical results recently identifying algebras over certain operads [4, 25, 17, 28, 16, 10, 14, 11]. See [1, 26] for expository surveys of the basics of operad theory. Before citing any of these results, let us mention some trivial classical examples. Let A denote one of the three words: “commutative”, “associative” and “Lie”. In each of these cases, consider the...

We establish several results regarding the algebra of derivations of tensor product of two algebras, and its connection to finite order automorphisms. These results generalize some well-know theorems in the literature. Dedicated to Professor Bruce Allison on the occasion of his sixtieth birthday 0. Introduction In 1969, R. E. Block [B] showed that the algebra of derivations of tensor product of...

We develop general results on centroids of Lie algebras and apply them to determine the centroid of extended affine Lie algebras, loop-like and Kac-Moody Lie algebras, and Lie algebras graded by finite root systems.

We investigate a class of Lie algebras which we call generalized reductive Lie algebras. These are generalizations of semi-simple, reductive, and affine Kac-Moody Lie algebras. A generalized reductive Lie algebra which has an irreducible root system is said to be irreducible and we note that this class of algebras have been under intensive investigation in recent years. They have also been call...

Introduction 2 1. Lie algebras: recollections 3 1.1. The basics 3 1.2. Scaling the structure 3 1.3. Filtrations 4 1.4. The Chevalley complex 4 1.5. The functor of primitives 6 1.6. The enhanced adjunction 6 1.7. The symmetric Hopf algebra 8 2. Looping Lie algebras 9 2.1. Group-Lie algebras 10 2.2. Forgetting to group structure 10 2.3. Chevalley complex of group-Lie algebras 11 2.4. Chevalley co...

In this dissertation, we investigate the cohomology theory of restricted Lie algebras. Motivations for the definition of a restricted Lie algebra are given and the theory of ordinary Lie algebra cohomology is briefly reviewed, including a discussion on algebraic interpretations of the low dimensional cohomology spaces of ordinary Lie algebras. The general Cartan-Eilenberg construction of the st...