We explicitly construct a particular real form of the Lie algebra in terms of symplectic matrices over the octonions, thus justifying the identifications and, at the group level, . Along the way, we provide a geometric description of the minimal representation of in terms of rank 3 objects called cubies.
We show by explicit calculations in the particular case of the 4-dimensional irreducible representation of that it is not always possible to generalize to the quantum case the notion of symmetric algebra of a Lie algebra representation.
We classify the -dimensional compact nilmanifolds that admit abelian complex structures, and for any such complex structure we describe the space of symplectic forms which are compatible with .
Let be a simple Lie algebra and the poset of non-trivial abelian ideals of a fixed Borel subalgebra of . In [8], we constructed a partition parameterised by the long positive roots of and studied the subposets . In this note, we show that this partition is compatible with intersections, relate it to the Kostant-Peterson parameterisation and to the centralisers of abelian ideals. We also prove that the poset of positive roots of is a join-semilattice.
The knowledge of the natural graded algebras of a given class of Lie algebras offers essential information about the structure of the class. So far, the classification of naturally graded Lie algebras is only known for some families of p-filiform Lie algebras. In certain sense, if g is a naturally graded Lie algebra of dimension n, the first case of no p-filiform Lie algebras it happens when the characteristic sequence is (n-3,2,1). We present the classification of a particular family of these algebras...
In this paper we construct and study an action of the affine braid group associated with a semi-simple algebraic group on derived categories of coherent sheaves on various varieties related to the Springer resolution of the nilpotent cone. In particular, we describe explicitly the action of the Artin braid group. This action is a “categorical version” of Kazhdan-Lusztig-Ginzburg’s construction of the affine Hecke algebra, and is used in particular by the first author and I. Mirković in the course...