Solvable groups, free divisors and nonisolated matrix singularities. I: Towers of free divisors. (Groupes solvables, diviseurs libres et singularités des matrices nonisolées I : Tours des diviseurs libres.) (English. French summary) Zbl 1371.14056
Summary: We introduce a method for obtaining new classes of free divisors from representations \(V\) of connected linear algebraic groups \(G\) where \(\dim G = \dim V\), with \(V\) having an open orbit. We give sufficient conditions that the complement of this open orbit, the “exceptional orbit variety”, is a free divisor (or a slightly weaker free* divisor) for “block representations” of both solvable groups and extensions of reductive groups by them. These are representations for which the matrix defined from a basis of associated “representation vector fields” on \(V\) has block triangular form, with blocks satisfying certain nonsingularity conditions.{
} For towers of Lie groups and representations this yields a tower of free divisors, successively obtained by adjoining varieties of singular matrices. This applies to solvable groups which give classical Cholesky-type factorization, and a modified form of it, on spaces of \(m \times m\) symmetric, skew-symmetric or general matrices. For skew-symmetric matrices, it further extends to representations of nonlinear infinite dimensional solvable Lie algebras.
For Part II, see Geom. Topol. 18, No. 2, 911–962 (2014; Zbl 1301.32017).
For Part II, see Geom. Topol. 18, No. 2, 911–962 (2014; Zbl 1301.32017).
MSC:
| 14L35 | Classical groups (algebro-geometric aspects) |
| 11S90 | Prehomogeneous vector spaces |
| 17B66 | Lie algebras of vector fields and related (super) algebras |
| 20G05 | Representation theory for linear algebraic groups |
Keywords:
prehomogeneous vector spaces; free divisors; linear free divisors; determinantal varieties; Pfaffian varieties; solvable algebraic groups; Cholesky-type factorizations; block representations; exceptional orbit varieties; infinite-dimensional solvable Lie algebrasCitations:
Zbl 1301.32017References:
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