Vogan Diagrams Research Articles

Combinatorics of Vogan diagrams for almost-Kähler manifolds

Let [Formula: see text] be a non-compact classical semisimple Lie group and let [Formula: see text] be the adjoint orbit with respect to a fixed element in [Formula: see text]. These manifolds can be equipped with an almost-Kähler structure and we provide explicit formulae for the existence of special almost-complex structures on [Formula: see text] purely in terms of the combinatorics of the associated Vogan diagram. The formulae are given separately for Lie groups whose Lie algebras are of type [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], where [Formula: see text] denotes the rank of the Lie algebra.

Equal rank real forms of affine non-twisted Kac-Moody Lie superalgebras

We study the equal rank real forms of affine non-twisted Kac-Moody Lie superalgebras by Cartan automorphisms and Vogan diagrams. We introduce admissible positive root systems and Hermitian real forms, then show that a real form has admissible positive root system if and only if it is Hermitian. As a result, we use the Vogan diagrams to classify the Hermitian real forms.

Affine Vogan diagrams and symmetric pairs

We introduce the affine Vogan diagrams of complex simple Lie algebras. These are generalizations of Vogan diagrams, and we study the involutions represented by them. We apply these diagrams to study the symmetric pairs, in particular the associated and symplectic symmetric pairs.

Admissible positive systems of affine Kac–Moody Lie algebras: The twisted cases

We introduce admissible positive systems and Hermitian real forms of affine twisted Kac–Moody Lie algebras, and show that a real form has admissible positive system if and only if it is Hermitian. We use the Vogan diagrams to classify the Hermitian real forms, and show that their symmetric spaces carry complex structures. The affine non-twisted Kac–Moody Lie algebras have been treated in an earlier work, and this article deals with the twisted cases.

Admissible positive systems of affine non-twisted Kac–Moody Lie algebras

We introduce admissible positive systems and Hermitian real forms of affine non-twisted Kac–Moody Lie algebras, and show that a real form has admissible positive system if and only if it is Hermitian. We use the Vogan diagrams to classify the Hermitian real forms, and show that their symmetric spaces carry complex structures.

Isoparametric foliations on complex projective spaces

Irreducible isoparametric foliations of arbitrary codimension q q on complex projective spaces C P n \mathbb {C} P^n are classified, for ( q , n ) ≠ ( 1 , 15 ) (q,n)\neq (1,15) . Remarkably, there are noncongruent examples that pull back under the Hopf map to congruent foliations on the sphere. Moreover, there exist many inhomogeneous isoparametric foliations, even of higher codimension. In fact, every irreducible isoparametric foliation on C P n \mathbb {C} P^n is homogeneous if and only if n + 1 n+1 is prime. The main tool developed in this work is a method to study singular Riemannian foliations with closed leaves on complex projective spaces. This method is based on a certain graph that generalizes extended Vogan diagrams of inner symmetric spaces.

VOGAN DIAGRAMS OF AFFINE TWISTED LIE SUPERALGEBRAS

A Vogan diagram is a Dynkin diagram with a Cartan involution of twisted affine superlagebras based on maximally compact Cartan subalgebras. This article construct the Vogan diagrams of twisted affine superalgebras. AMS Subject Classification: 17B20, 17B22, 17B40

Cartan automorphisms and Vogan superdiagrams

The real forms of complex semisimple Lie algebras are characterized by Cartan involutions and Vogan diagrams. We extend these notions to the Cartan automorphisms and Vogan superdiagrams, and show that they characterize the real forms of complex contragredient Lie superalgebras.

Double Vogan diagrams and semisimple symmetric spaces

A Vogan diagram is a set of involution and painting on a Dynkin diagram. It selects a real form, or equivalently an involution, from a complex simple Lie algebra. We introduce the double Vogan diagram, which is two sets of Vogan diagrams superimposed on an affine Dynkin diagram. They correspond to pairs of commuting involutions on complex simple Lie algebras, and therefore provide an independent classification of the simple locally symmetric pairs.

Vogan diagrams of twisted affine Kac–Moody Lie algebras

A Vogan diagram is a Dynkin diagram of a Kac-Moody Lie algebra of finite or affine type overlayed with additional structures. This paper develops the theory of Vogan diagrams for almost compact real forms of indecomposable twisted affine Kac- Moody Lie algebras and shows that equivalence classes of Vogan diagrams correspond to isomorphism classes of almost compact real forms of twisted affine Kac-Moody Lie algebras as given by H. Ben Messaoud and G. Rousseau.

A quick proof on the equivalence classes of extended Vogan diagrams

An extended Vogan diagram is an extended Dynkin diagram together with a diagram involution, such that the vertices fixed by the involution are colored white or black. Every extended Vogan diagram represents an almost compact real form of the affine Kac–Moody Lie algebra. Two extended diagrams are said to be equivalent if they represent isomorphic real forms. The equivalence classes of extended Vogan diagrams have earlier been classified by the authors. In this paper, we present a much shorter and instructive argument.

Extended Vogan diagrams

An extended Vogan diagram is an extended Dynkin diagram with a diagram involution, such that the vertices fixed by the involution can be painted or unpainted. Every extended Vogan diagram represents an almost compact real form of some affine Kac–Moody Lie algebra. Two diagrams may represent isomorphic algebras, and in this case we say that the diagrams are equivalent. In this paper, we classify the equivalence classes of extended Vogan diagrams, and provide a complete list of all diagrams within each class. It gives a combinatorial classification of the isomorphic classes of almost compact real forms of the affine Kac–Moody Lie algebras.

Equivalence classes of Vogan diagrams

A Vogan diagram is a Dynkin diagram with an involution, and the vertices fixed by the involution may be painted. They represent real simple Lie algebras, and two diagrams are said to be equivalent if they represent the same Lie algebra. In this article we classify the equivalence classes of all Vogan diagrams. In doing so, we find that the underlying Dynkin diagrams have certain properties in graph painting. We show that this combinatorial property provides an easy classification for most of the simply-laced Dynkin diagrams.

Langlands parameters of derived functor modules and Vogan diagrams

Let $G$ be a linear reductive Lie group with finite center, let $K$ be a maximal compact subgroup, and assume that $\mathrm{rank } G = \mathrm {rank } K$. Let $\ g= l\oplus u$ be a $\theta$ stable parabolic subalgebra obtained by building $l$ from a subset of the compact simple roots and form $A_g(\lambda)$. Suppose $\Lambda=\lambda+2\delta( u\cap p)$ is $K$-dominant and the infinitesimal character, $\lambda+\delta$, of $A_{g}(\lambda)$ is nondominant due to a noncompact simple root. By interpreting these conditions on the level of Vogan diagrams, a conjecture by Knapp is (essentially) settled for the groups $G=SU(p,q),\, Sp(p,q)$, and $SO^*(2n)$, thereby determining the Langlands parameters of natural irreducible subquotient of $A_{ g}(\lambda)$. For the remaining classical groups, simple supplementary conditions are given under which the Langlands parameters may be determined.

Vogan Diagrams of Real Forms of Affine Kac–Moody Lie Algebras

A Vogan diagram is actually a Dynkin diagram with some additional structure. This paper develops theory of Vogan diagrams for “almost compact” real forms of indecomposable nontwisted affine Kac–Moody Lie algebras. Here, the equivalence classes of Vogan diagrams are in one–one correspondence with the isomorphism classes of almost compact real forms.

Invariants of Real Forms of Affine Kac-Moody Lie Algebras

The theory of Vogan diagrams, which are Dynkin diagrams with an overlay of certain additional information, allows one to give a rapid classification of finite-dimensional real semisimple Lie algebras and to make use of this classification in practice. This paper develops a corresponding theory of Vogan diagrams for “almost compact” real forms of indecomposable nontwisted affine Kac–Moody Lie algebras. In this case also, the equivalence classes of Vogan diagrams correspond to the isomorphism classes of almost compact real forms. Although the real forms of such algebras had already been classified, the theory of Vogan diagrams introduces invariants for such algebras and makes it possible to locate a given real form within the classification.