Decomposition Varieties in Semisimple Lie Algebras

1998 ◽  
Vol 50 (5) ◽  
pp. 929-971 ◽  
Author(s):  
Abraham Broer
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Nilpotent Orbits ◽  
Semisimple Lie Algebra ◽  
Semisimple Lie Algebras ◽  
Rational Singularities ◽  
Common Generalization ◽  
Simultaneous Resolution ◽  
Decomposition Class

AbstractThe notion of decompositon class in a semisimple Lie algebra is a common generalization of nilpotent orbits and the set of regular semisimple elements.We prove that the closure of a decomposition class has many properties in common with nilpotent varieties, e.g., its normalization has rational singularities.The famous Grothendieck simultaneous resolution is related to the decomposition class of regular semisimple elements. We study the properties of the analogous commutative diagrams associated to an arbitrary decomposition class.

scholarly journals KRULL DIMENSION OF AFFINOID ENVELOPING ALGEBRAS OF SEMISIMPLE LIE ALGEBRAS

2013 ◽  
Vol 55 (A) ◽  
pp. 7-26
Author(s):  
KONSTANTIN ARDAKOV ◽  
IAN GROJNOWSKI
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Krull Dimension ◽  
Semisimple Lie Algebra ◽  
Semisimple Lie Algebras ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Complex Semisimple

AbstractUsing Beilinson–Bernstein localisation, we give another proof of Levasseur's theorem on the Krull dimension of the enveloping algebra of a complex semisimple Lie algebra. The proof also extends to the case of affinoid enveloping algebras.

scholarly journals Rings with involution whose symmetric elements are central

1980 ◽  
Vol 3 (2) ◽  
pp. 247-253
Author(s):  
Taw Pin Lim
Keyword(s):  
Commutative Ring ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Semisimple Lie Algebra ◽  
Direct Sum ◽  
Rings With Involution ◽  
Torsion Free

In a ringRwith involution whose symmetric elementsSare central, the skew-symmetric elementsKform a Lie algebra over the commutative ringS. The classification of such rings which are2-torsion free is equivalent to the classification of Lie algebrasKoverSequipped with a bilinear formfthat is symmetric, invariant and satisfies[[x,y],z]=f(y,z)x−f(z,x)y. IfSis a field of char≠2,f≠0anddimK>1thenKis a semisimple Lie algebra if and only iffis nondegenerate. Moreover, the derived algebraK′is either the pure quaternions overSor a direct sum of mutually orthogonal abelian Lie ideals ofdim≤2.

scholarly journals Computing with Nilpotent Orbits in Simple Lie Algebras of Exceptional Type

2008 ◽  
Vol 11 ◽  
pp. 280-297 ◽  
Author(s):  
Willem A. de Graaf
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Simple Algorithm ◽  
Adjoint Action ◽  
Closed Field ◽  
Nilpotent Orbits ◽  
Simple Algebraic Group ◽  
Algebraically Closed Field ◽  
Simple Lie Algebras

AbstractLet G be a simple algebraic group over an algebraically closed field with Lie algebra g. Then the orbits of nilpotent elements of g under the adjoint action of G have been classified. We describe a simple algorithm for finding a representative of a nilpotent orbit. We use this to compute lists of representatives of these orbits for the Lie algebras of exceptional type. Then we give two applications. The first one concerns settling a conjecture by Elashvili on the index of centralizers of nilpotent orbits, for the case where the Lie algebra is of exceptional type. The second deals with minimal dimensions of centralizers in centralizers.

ON SEMIREGULAR NILPOTENT ORBITS IN SIMPLE LIE ALGEBRAS OF TYPE D

2002 ◽  
Vol 01 (03) ◽  
pp. 341-356 ◽  
Author(s):  
BENOÎT ARBOUR ◽  
DRAGOMIR Ž. ĐOKOVIĆ
Keyword(s):  
Linear Combination ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Cartan Subalgebra ◽  
Nilpotent Orbits ◽  
Connected Component ◽  
Real Form ◽  
Simple Complex ◽  
Type D ◽  
Explicit Formulae

We derive explicit formulae for the characteristics H(k) of the semiregular nilpotent orbits Dn(ak) of the simple complex Lie algebra [Formula: see text] of type Dn. These formulae express H(k) as an integral linear combination of a basis of the Cartan subalgebra [Formula: see text] of [Formula: see text]. For that purpose we use several suitable bases of [Formula: see text] consisting of coroots. We also construct several explicit standard triples (E, H, F) with H = H(k), and E, F ∈ Dn(ak). Similar triples are constructed also for each connected component of the intersection of the orbit Dn(ak) with the split real form [Formula: see text] and the real form [Formula: see text] of [Formula: see text].

scholarly journals Constructions of Representations of Rank Two Semisimple Lie Algebras with Distributive Lattices

10.37236/1135 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
L. Wyatt Alverson II ◽  
Robert G. Donnelly ◽  
Scott J. Lewis ◽  
Robert Pervine
Keyword(s):  
Irreducible Representation ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Irreducible Representations ◽  
Distributive Lattices ◽  
Semisimple Lie Algebra ◽  
Order Ideals ◽  
Symplectic Representations ◽  
Extremal Properties

We associate one or two posets (which we call "semistandard posets") to any given irreducible representation of a rank two semisimple Lie algebra over ${\Bbb C}$. Elsewhere we have shown how the distributive lattices of order ideals taken from semistandard posets (we call these "semistandard lattices") can be used to obtain certain information about these irreducible representations. Here we show that some of these semistandard lattices can be used to present explicit actions of Lie algebra generators on weight bases (Theorem 5.1), which implies these particular semistandard lattices are supporting graphs. Our descriptions of these actions are explicit in the sense that relative to the bases obtained, the entries for the representing matrices of certain Lie algebra generators are rational coefficients we assign in pairs to the lattice edges. In Theorem 4.4 we show that if such coefficients can be assigned to the edges, then the assignment is unique up to products; we conclude that the associated weight bases enjoy certain uniqueness and extremal properties (the "solitary" and "edge-minimal" properties respectively). Our proof of this result is uniform and combinatorial in that it depends only on certain properties possessed by all semistandard posets. For certain families of semistandard lattices some of these results were obtained in previous papers; in Proposition 5.6 we explicitly construct new weight bases for a certain family of rank two symplectic representations. These results are used to help obtain in Theorem 5.1 the classification of those semistandard lattices which are supporting graphs.

Nilpotent Orbits in Semisimple Lie Algebras

2017 ◽  
Author(s):  
David H. Collingwood ◽  
William M. McGovern
Keyword(s):  
Lie Algebras ◽  
Nilpotent Orbits ◽  
Semisimple Lie Algebras

scholarly journals ON A FORMULA FOR THE PI-EXPONENT OF LIE ALGEBRAS

2013 ◽  
Vol 13 (01) ◽  
pp. 1350069 ◽  
Author(s):  
A. S. GORDIENKO
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Simple Formula ◽  
Natural Generalization ◽  
Rational Action ◽  
Semisimple Lie Algebra ◽  
Arbitrary Group ◽  
Solvable Radical ◽  
Finite Dimensional ◽  
Algebra Over A Field

We prove that one of the conditions in Zaicev's formula for the PI-exponent and in its natural generalization for the Hopf PI-exponent, can be weakened. Using the modification of the formula, we prove that if a finite-dimensional semisimple Lie algebra acts by derivations on a finite-dimensional Lie algebra over a field of characteristic 0, then the differential PI-exponent coincides with the ordinary one. Analogously, the exponent of polynomial G-identities of a finite-dimensional Lie algebra with a rational action of a connected reductive affine algebraic group G by automorphisms, coincides with the ordinary PI-exponent. In addition, we provide a simple formula for the Hopf PI-exponent and prove the existence of the Hopf PI-exponent itself for H-module Lie algebras whose solvable radical is nilpotent, assuming only the H-invariance of the radical, i.e. under weaker assumptions on the H-action, than in the general case. As a consequence, we show that the analog of Amitsur's conjecture holds for G-codimensions of all finite-dimensional Lie G-algebras whose solvable radical is nilpotent, for an arbitrary group G.

Sign in / Sign up

Export Citation Format

Share Document