Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fomenko-Mischenko Theory, Hessenberg Varieties, and Polarizations   The symmetric algebra g (denoted S(\g)) over a Lie algebra \g (frak g) has
the structure of a Poisson algebra. Assume \g is complex semi-simple. Then
results of Fomenko- Mischenko (translation of invariants) and A.Tarasev
construct a polynomial subalgebra \cal H = \bf C[q_1,...,q_b] of S(\g) which is
maximally Poisson commutative. Here b is the dimension of a Borel subalgebra of
\g. Let G be the adjoint group of \g and let \ell = rank \g. Identify \g with
its dual so that any G-orbit O in \g has the structure (KKS) of a symplectic
manifold and S(\g) can be identified with the affine algebra of \g. An element
x \in \g is strongly regular if \{(dq_i)_x\}, i=1,...,b, are linearly
independent. Then the set \g^{sreg} of all strongly regular elements is Zariski
open and dense in \g, and also \g^{sreg \subset \g^{reg} where \g^{reg} is the
set of all regular elements in \g. A Hessenberg variety is the b-dimensional
affine plane in \g, obtained by translating a Borel subalgebra by a suitable
principal nilpotent element. This variety was introduced in [K2]. Defining Hess
to be a particular Hessenberg variety, Tarasev has shown that Hess \subset
\g^sreg. Let R be the set of all regular G-orbits in \g. Thus if O \in R, then
O is a symplectic manifold of dim 2n where n= b-\ell. For any O\in R let
O^{sreg} = \g^{sreg}\cap O. We show that O^{sreg} is Zariski open and dense in
O so that O^{sreg} is again a symplectic manifold of dim 2n. For any O \in R
let Hess (O) = Hess \cap O. We prove that Hess(O) is a Lagrangian submanifold
of O^{sreg} and Hess =\sqcup_{O \in R} Hess(O). The main result here shows that
there exists, simultaneously over all O \in R, an explicit polarization (i.e.,
a "fibration" by Lagrangian submanifolds) of O^{sreg} which makes O^{sreg}
simulate, in some sense, the cotangent bundle of Hess(O).

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The categories of this paper are: acc-phys
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The categories of this paper are: plasm-ph
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The categories of this paper are: eess
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The categories of this paper are: chao-dyn
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The categories of this paper are: gr-qc
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The categories of this paper are: ao-sci
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The categories of this paper are: astro-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: mtrl-th
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The categories of this paper are: cond-mat
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The categories of this paper are: q-bio
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The categories of this paper are: alg-geom
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The categories of this paper are: q-fin
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The categories of this paper are: patt-sol
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-lat