The quotient of the adjoint action

1980 ◽  
pp. 17-42
Author(s):  
Peter Slodowy
Keyword(s):  
Adjoint Action

scholarly journals Geometric aspects of self-adjoint Sturm–Liouville problems

2017 ◽  
Vol 147 (6) ◽  
pp. 1279-1295
Author(s):  
Yicao Wang
Keyword(s):  
Boundary Conditions ◽  
Liouville Equation ◽  
Adjoint Action ◽  
Principal Type ◽  
Unitary Matrices ◽  
Adjoint Orbits ◽  
Sturm Liouville ◽  
Refined Classification ◽  
Adjoint Orbit

In this paper we use U(2), the group of 2 × 2 unitary matrices, to parametrize the space of all self-adjoint boundary conditions for a fixed Sturm–Liouville equation on the interval [0, 1]. The adjoint action of U(2) on itself naturally leads to a refined classification of self-adjoint boundary conditions – each adjoint orbit is a subclass of these boundary conditions. We give explicit parametrizations of those adjoint orbits of principal type, i.e. orbits diffeomorphic to the 2-sphere S2, and investigate the behaviour of the nth eigenvalue λnas a function on such orbits.

scholarly journals Distance to the convex hull of an orbit under the action of a compact Lie group

1999 ◽  
Vol 66 (3) ◽  
pp. 331-357 ◽  
Author(s):  
Randall R. Holmes ◽  
Tin-Yau Tam
Keyword(s):  
Convex Hull ◽  
Analogous Result ◽  
Maximal Compact Subgroup ◽  
Reductive Group ◽  
Compact Subgroup ◽  
Adjoint Action ◽  
Reflection Group ◽  
Compact Lie Group ◽  
Real Vector ◽  
Finite Reflection Group

AbstractFor a real vector space V acted on by a group K and fixed x and y in V, we consider the problem of finding the minimum (respectively, maximum) distance, relative to a K-invariant convex function on V, between x and elements of the convex hull of the K-orbit of y. We solve this problem in the case where V is a Euclidean space and K is a finite reflection group acting on V. Then we use this result to obtain an analogous result in the case where K is a maximal compact subgroup of a reductive group G with adjoint action on the vector component ρ of a Cartan decomposition of Lie G. Our results generalize results of Li and Tsing and of Cheng concerning distances to the convex hulls of matrix orbits.

Classes of unipotent elements in simple algebraic groups. I

1976 ◽  
Vol 79 (3) ◽  
pp. 401-425 ◽  
Author(s):  
P. Bala ◽  
R. W. Carter
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Algebraic Groups ◽  
Adjoint Action ◽  
Conjugacy Classes ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Nilpotent Elements

LetGbe a simple adjoint algebraic group over an algebraically closed fieldK. We are concerned to describe the conjugacy classes of unipotent elements ofG. Goperates on its Lie algebra g by means of the adjoint action and we may consider classes of nilpotent elements of g under this action. It has been shown by Springer (11) that there is a bijection between the unipotent elements ofGand the nilpotent elements ofgwhich preserves theG-action, provided that the characteristic ofKis either 0 or a ‘good prime’ forG. Thus we may concentrate on the problem of classifying the nilpotent elements of g under the adjointG-action.

scholarly journals LOCAL NATURE OF COSET MODELS

2004 ◽  
Vol 16 (03) ◽  
pp. 353-382 ◽  
Author(s):  
S. KÖSTER
Keyword(s):  
Representation Formula ◽  
Adjoint Action ◽  
Energy Tensor ◽  
Local Algebras ◽  
Relative Commutants ◽  
Stress Energy ◽  
Local Observables ◽  
Coset Models ◽  
Geometric Picture ◽  
Local Nature

The local algebras of the maximal Coset model [Formula: see text] associated with a chiral conformal subtheory [Formula: see text] are shown to coincide with the local relative commutants of [Formula: see text] in ℬ, provided [Formula: see text] possesses a stress-energy tensor.Making the same assumption, the adjoint action of the unique inner-implementing representation [Formula: see text] associated with [Formula: see text] on the local observables in ℬ is found to define net-endomorphisms of ℬ. This property is exploited for constructing from ℬ a conformally covariant holographic image in (1+1) dimensions which proves useful as a geometric picture for the joint inclusion [Formula: see text].Immediate applications to the analysis of current subalgebras are given and the relation to normal canonical tensor product subfactors is clarified. A natural converse of Borchers' theorem on half-sided translations is made accessible.

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.

scholarly journals The Monoidal Category of Yetter-Drinfeld Modules over a Weak Braided Hopf Algebra

2015 ◽  
Vol 22 (spec01) ◽  
pp. 871-902
Author(s):  
J.N. Alonso Álvarez ◽  
J.M. Fernández Vilaboa ◽  
R. González Rodríguez ◽  
C. Soneira Calvo
Keyword(s):  
Hopf Algebra ◽  
Monoidal Category ◽  
Adjoint Action ◽  
Drinfeld Modules ◽  
Braided Hopf Algebra

In this paper we introduce the notion of weak operators and the theory of Yetter-Drinfeld modules over a weak braided Hopf algebra with invertible antipode in a strict monoidal category. We prove that the class of such objects constitutes a non-strict monoidal category. It is also shown that this category is not trivial, that is to say, it admits objects generated by the adjoint action (coaction) associated to the weak braided Hopf algebra.

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