Groupes et algébres de lie

Solutions of the classical Yang - Baxter equation for simple Lie algebras

The coadjoint orbits of the Virasoro group, which have been investigated by Lazutkin and Pankratova and by Segal, should according to the Kirillov-Kostant theory be related to the unitary representations of the Virasoro group. In this paper, the classification of orbits is reconsidered, with an explicit description of the possible centralizers of coadjoint orbits. The possible orbits are diff(S1) itself, diff(S1)/S1, and diff(S1)/SL(n)(2,R), withSL(n)(2,R) a certain discrete series of embeddings ofSL(2,R) in diff(S1), and diffS1/T, whereT may be any of certain rather special one parameter subgroups of diffS1. An attempt is made to clarify the relation between orbits and representations. It appears that quantization of diffS1/S1 is related to unitary representations with nondegenerate Kac determinant (unitary Verma modules), while quantization of diffS1/SL(n)(2,R) is seemingly related to unitary representations with null vectors in leveln. A better understanding of how to quantize the relevant orbits might lead to a better geometrical understanding of Virasoro representation theory. In the process of investigating Virasoro coadjoint orbits, we observe the existence of left invariant symplectic structures on the Virasoro group manifold. As is described in an appendix, these give rise to Lie bialgebra structures with the Virasoro algebra as the underlying Lie algebra.

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Coadjoint Orbits of the Virasoro Group

On frobenius lie algebras

The paper contains a classification of the three-dimensional real and complex Lie bialgebras, with a description of the corresponding Drinfel’d doubles.

Classification of three-dimensional Lie bialgebras

LIMBURGS UNIV CENTRUM,DEPT WNI,B-3500 DIEPENBEEK,BELGIUM.Alev, J, UNIV REIMS,UFR SCI,DEPT MATH,MOULIN HOUSSE,BP 347,F-51062 REIMS,FRANCE.

https://www.ams.org/tran/1996-348-05/S0002-9947-96-01465-1/S0002-9947-96-01465-1.pdf

Alfons I. Ooms

Papers

On frobenius lie algebras

A class of counterexamples to the Gel’fand-Kirillov conjecture

A degree inequality for lie algebras with a regular poisson semi-center

Computing invariants and semi-invariants by means of Frobenius Lie algebras

The Gelfand-Kirillov conjecture for Lie algebras of dimension at most eight