524 research outputs found
Orthogonal Decomposition of Some Affine Lie Algebras in Terms of their Heisenberg Subalgebras
In the present note we suggest an affinization of a theorem by Kostrikin
et.al. about the decomposition of some complex simple Lie algebras
into the algebraic sum of pairwise orthogonal Cartan subalgebras. We point out
that the untwisted affine Kac-Moody algebras of types ( prime,
), can be decomposed into
the algebraic sum of pairwise or\-tho\-go\-nal Heisenberg subalgebras. The
and cases are discussed in great detail. Some possible
applications of such decompositions are also discussed.Comment: 16 pages, LaTeX, no figure
The variety of reductions for a reductive symmetric pair
We define and study the variety of reductions for a reductive symmetric pair
(G,theta), which is the natural compactification of the set of the Cartan
subspaces of the symmetric pair. These varieties generalize the varieties of
reductions for the Severi varieties studied by Iliev and Manivel, which are
Fano varieties.
We develop a theoretical basis to the study these varieties of reductions,
and relate the geometry of these variety to some problems in representation
theory. A very useful result is the rigidity of semi-simple elements in
deformations of algebraic subalgebras of Lie algebras.
We apply this theory to the study of other varieties of reductions in a
companion paper, which yields two new Fano varieties.Comment: 23 page
- …