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Equations of some wonderful compactifications

Pascal Hivert (2011)

Annales de l’institut Fourier

De Concini and Procesi have defined the wonderful compactification X ¯ of a symmetric space X = G / G σ where G is a complex semisimple adjoint group and G σ the subgroup of fixed points of G by an involution σ . It is a closed subvariety of a Grassmannian of the Lie algebra 𝔤 of G . In this paper we prove that, when the rank of X is equal to the rank of G , the variety is defined by linear equations. The set of equations expresses the fact that the invariant alternate trilinear form w on 𝔤 vanishes on the ( - 1 ) -eigenspace...

Equivalence of control systems with linear systems on Lie groups and homogeneous spaces

Philippe Jouan (2010)

ESAIM: Control, Optimisation and Calculus of Variations

The aim of this paper is to prove that a control affine system on a manifold is equivalent by diffeomorphism to a linear system on a Lie group or a homogeneous space if and only if the vector fields of the system are complete and generate a finite dimensional Lie algebra. A vector field on a connected Lie group is linear if its flow is a one parameter group of automorphisms. An affine vector field is obtained by adding a left invariant one. Its projection on a homogeneous space, whenever it exists,...

Equivariant deformation quantization for the cotangent bundle of a flag manifold

Ranee Brylinski (2002)

Annales de l’institut Fourier

Let X be a (generalized) flag manifold of a complex semisimple Lie group G . We investigate the problem of constructing a graded star product on ℛ = R ( T ☆ X ) which corresponds to a G -equivariant quantization of symbols into twisted differential operators acting on half-forms on X . We construct, when ℛ is generated by the momentum functions μ x for G , a preferred choice of ☆ where μ x ☆ φ has the form μ x φ + 1 2 { μ x , φ } t + Λ x ( φ ) t 2 . Here Λ x are operators on ℛ . In the known examples, Λ x ( x ≠ 0 ) is not a differential operator, and so the star product μ x ☆ φ ...

Equivariant Morita equivalences between Podleś spheres

Kenny De Commer (2012)

Banach Center Publications

We show that the family of Podleś spheres is complete under equivariant Morita equivalence (with respect to the action of quantum SU(2)), and determine the associated orbits. We also give explicit formulas for the actions which are equivariantly Morita equivalent with the quantum projective plane. In both cases, the computations are made by examining the localized spectral decomposition of a generalized Casimir element.

Euclidean components for a class of self-injective algebras

Sarah Scherotzke (2009)

Colloquium Mathematicae

We determine the length of composition series of projective modules of G-transitive algebras with an Auslander-Reiten component of Euclidean tree class. We thereby correct and generalize a result of Farnsteiner [Math. Nachr. 202 (1999)]. Furthermore we show that modules with certain length of composition series are periodic. We apply these results to G-transitive blocks of the universal enveloping algebras of restricted p-Lie algebras and prove that G-transitive principal blocks only allow components...

Eulerian idempotent and Kashiwara-Vergne conjecture

Emily Burgunder (2008)

Annales de l’institut Fourier

By using the interplay between the Eulerian idempotent and the Dynkin idempotent, we construct explicitly a particular symmetric solution ( F , G ) of the first equation of the Kashiwara-Vergne conjecture x + y - log ( e y e x ) = ( 1 - e - ad x ) F ( x , y ) + ( e ad y - 1 ) G ( x , y ) . Then, we explicit all the solutions of the equation in the completion of the free Lie algebra generated by two indeterminates x and y thanks to the kernel of the Dynkin idempotent.

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