1. Lie groups and Lie algebras 2 1.1. Definition and examples of Lie groups 2 1.2. Left invariant vector fields and Lie algebras 3 1.3. The exponential map 6 2. Representations of Lie groups and Lie algebras 9 2.1. Definitions and examples 9 2.2. The adjoint representation 9 2.3. Semisimplicity and Schur’s lemma 9 2.4. Haar measure and Weyl’s trick 9 2.5. Characters 9 2.6. Examples 9 2.7. Peter...

We present a systematic procedure for the determination of a complete set of kth-order (k ≥ 2) differential invariants corresponding to vector fields in three variables for three-dimensional Lie algebras. In addition, we give a procedure for the construction of a system of two kth-order ODEs admitting three-dimensional Lie algebras from the associated complete set of invariants and show that th...

We explore a relationship between the classical representation theory of a complex, semisimple Lie algebra g and the resonance varieties R(V,K) ⊂ V ∗ attached to irreducible g-modules V and submodules K ⊂ V ∧ V . In the process, we give a precise roots-and-weights criterion insuring the vanishing of these varieties, or, equivalently, the finiteness of certain modules W(V,K) over the symmetric a...

I gave two talks on some of my results on toroidal Lie algebras in the conference Functional Analysis VIII held in Dubrovnik, Croatia in June 2003. But these results have been submitted to research journals and will appear soon. So I decided to write an expository article on Toroidal Lie Algebras covering my results for the proceedings of Functional Analysis VIII. I will introduce the definitio...

Linear codes with large minimal distances are important error correcting codes in information theory. Orthogonal codes have more applications in the other fields of mathematics. In this paper, we study the binary and ternary orthogonal codes generated by the weight matrices on finite-dimensional modules of simple Lie algebras. The Weyl groups of the Lie algebras act on these codes isometrically...

In spite of many efforts over the past 50 years, the irreducible representations of the Lie algebra of a simple algebraic group over a field of prime characteristic are poorly understood. Recent work on quantum groups at a root of unity has provided new impetus for the subject. This article surveys what has been done and what remains to be done.

1. Introduction. Let © be a Lie algebra over a field A. A decomposition of © is a triple (ni, b, n2) of subalgebras of © such that ® = niffif)©n2 a vector space direct sum and such that [b, n,-]Cn< for 4 = 1, 2. In [5] we showed how one could "induce" ©-modules from b-modules in a natural manner. In this paper we prove a useable necessary and sufficient condition that a ©-module must satisfy in...

We prove an explicit formula for the invariant μ(g) for finite-dimensional semisimple, and reductive Lie algebras g over C. Here μ(g) is the minimal dimension of a faithful linear representation of g. The result can be used to study Dynkin’s classification of maximal reductive subalgebras of semisimple Lie algebras.

Tensor products of irreducible representations of the Jordanian quantum algebras Uh(sl(2)) and Uh(su(1, 1)) are considered. For both the highest weight finite dimensional representations of Uh(sl(2)) and lowest weight infinite dimensional ones of Uh(su(1, 1)) , it is shown that tensor product representations are reducible and that the decomposition rules to irreducible representations are exact...

Weyl’s character formula is a useful tool in understanding the structure of irreducible representations of semisimple Lie algebras. Even more important than the character formula itself is a corollary, the Weyl dimension formula, which gives a fairly simple expression for the total dimension of the representation in terms of the root system of the Lie algebra. Although the results discussed are...