Classification of semisimple Lie algebras
ثبت نشده
چکیده
Furthermore h was diagonalisable in every irreducible representation and H := Span(h) is obviously an abelian subalgebra. Note that h = h + 0 is the abstract Jordan decomposition of h, that H = CL(H) is the weight space of H , acting on L with the adjoint action, corresponding to the weight 0 ∈ H . Likewise, Span(e) is the weight space for the weight c · h 7→ −2c for c ∈ C, and Span( f ) is the weight space for the weight c · h 7→ 2c for c ∈ C. This approach can be generalised. Our big plan will be:
منابع مشابه
Lecture 5: Semisimple Lie Algebras over C
In this lecture I will explain the classification of finite dimensional semisimple Lie algebras over C. Semisimple Lie algebras are defined similarly to semisimple finite dimensional associative algebras but are far more interesting and rich. The classification reduces to that of simple Lie algebras (i.e., Lie algebras with non-zero bracket and no proper ideals). The classification (initially d...
متن کاملClassification of Finite-dimensional Semisimple Lie Algebras
Every finite-dimensional Lie algebra is a semi-direct product of a solvable Lie algebra and a semisimple Lie algebra. Classifying the solvable Lie algebras is difficult, but the semisimple Lie algebras have a relatively easy classification. We discuss in some detail how the representation theory of the particular Lie algebra sl2 tightly controls the structure of general semisimple Lie algebras,...
متن کاملComplex Lie Algebras
We prove that every Lie algebra can be decomposed into a solvable Lie algebra and a semisimple Lie algebra. Then we show that every complex semisimple Lie algebra is a direct sum of simple Lie algebras. Finally, we give a complete classification of simple complex Lie algebras.
متن کاملLie algebras and the classification of semisimple algebraic groups
The Lie algebra of an algebraic group is the (first) linear approximation to the group. The study of Lie algebras is much more elementary than that of algebraic groups. For example, most of the results on Lie algebras that we shall need are proved already in the undergraduate text Erdmann and Wildon 2006. After many preliminaries, in 7 we describe the structure and classification of the semisi...
متن کاملSemisimple Algebraic Groups in Characteristic Zero
It is shown that the classification theorems for semisimple algebraic groups in characteristic zero can be derived quite simply and naturally from the corresponding theorems for Lie algebras by using a little of the theory of tensor categories. This article is extracted from Milne 2007. Introduction The classical approach to classifying the semisimple algebraic groups over C (see Borel 1975, §1...
متن کاملLecture 6: Kac-moody Algebras, Reductive Groups, and Representations
We start by introducing Kac-Moody algebras and completing the classification of finite dimensional semisimple Lie algebras. We then discuss the classification of finite dimensional representations of semisimple Lie algebras (and, more generally, integrable highest weight representations of Kac-Moody algebras). We finish by discussing the structure and representation theory of reductive algebrai...
متن کامل