2019/07/09 by Peter Crooks, Crooks, Peter, Markus Röser +1
Mathematics · #17B63 #17B80 (primary) #22E46 (secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT) #Symplectic Geometry (math.SG) #math.AG #math.RT #math.SG #msc:17B63 #msc:17B80 #msc:22E46
paper · pdf · doi:10.48550/arxiv.1907.04429
29 pages
arxiv created 2019/07/09 · arxiv updated 2019/07/11
This work is concerned with Mishchenko and Fomenko's celebrated theory of completely integrable systems on a complex semisimple Lie algebra \mathfrakg. Their theory associates a maximal Poisson-commutative subalgebra of ℂ[\mathfrakg] to each regular element a∈\mathfrakg, and one can assemble free generators of this subalgebra into a moment map Fa:\mathfrakg→ℂb. We examine the structure of fibres in Mishchenko--Fomenko systems, building on the foundation laid by Bolsinov, Charbonnel--Moreau, Moreau, and others. This includes proving that the critical values of Fa have codimension 1 or 2 in ℂb, and that each codimension is achievable in examples. Our results on singularities make use of a subalgebra \mathfrakba⊆\mathfrakg, defined to be the intersection of all Borel subalgebras of \mathfrakg containing a. In the case of a non-nilpotent a∈\mathfrakgreg and an element x∈\mathfrakba, we prove the following: x+[\mathfrakba,\mathfrakba] lies in the singular locus of Fa-1(Fa(x)), and the fibres through points in \mathfrakba form a rank(\mathfrakg)-dimensional family of singular fibres. We next consider the irreducible components of our fibres, giving a systematic way to construct many components via Mishchenko--Fomenko systems on Levi subalgebras \mathfrakl⊆\mathfrakg. In addition, we obtain concrete results on irreducible components that do not arise from the aforementioned construction. Our final main result is a recursive formula for the number of irreducible components in Fa-1(0), and it generalizes a result of Charbonnel--Moreau. Illustrative examples are included at the end of this paper.