Filtrations in Semisimple Lie Algebras, Ii
نویسندگان
چکیده
In this paper, we continue our study of the maximal bounded Z-filtrations of a complex semisimple Lie algebra L. Specifically, we discuss the functionals which give rise to such filtrations, and we show that they are related to certain semisimple subalgebras of L of full rank. In this way, we determine the “order” of these functionals and count them without the aid of computer computations. The main results here involve the Lie algebras of type E6, E7 and E8, since we already know a good deal about the functionals for the remaining types. Nevertheless, we reinterpret our previous results into the new context considered here. Finally, we describe the associated graded Lie algebras of all of the maximal filtrations obtained in this manner.
منابع مشابه
Filtrations in Semisimple Lie Algebras, I
In this paper, we study the maximal bounded Z-filtrations of a complex semisimple Lie algebra L. Specifically, we show that if L is simple of classical type An, Bn, Cn or Dn, then these filtrations correspond uniquely to a precise set of linear functionals on its root space. We obtain partial, but not definitive, results in this direction for the remaining exceptional algebras. Maximal bounded ...
متن کاملFiltrations in Semisimple Lie Algebras, III
This is the third in a series of papers. The first two, by Yiftach Barnea and this author, study the maximal bounded Z-filtrations of the finitedimensional simple Lie algebras over the complex numbers. Those papers obtain a complete characterization for all but the five exceptional Lie algebras, namely the ones of type G2, F4, E6, E7 and E8. Here, we fill in the missing step for the algebra G2....
متن کاملLecture 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 کاملNon-solvable contractions of semisimple Lie algebras in low dimension
The problem of non-solvable contractions of Lie algebras is analyzed. By means of a stability theorem, the problem is shown to be deeply related to the embeddings among semisimple Lie algebras and the resulting branching rules for representations. With this procedure, we determine all deformations of indecomposable Lie algebras having a nontrivial Levi decomposition onto semisimple Lie algebras...
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 کامل