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The Serre spectral sequence of a Lie subalgebroid
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We study a spectral sequence approximating Lie algebroid cohomology associated to a Lie subalgebroid. This is a simultaneous generalisation of several classical constructions in differential geometry, including the Leray-Serre spectral sequence for de Rham cohomology associated to a fibration, the Hochschild-Serre spectral sequence for Lie algebras, and the Mackenzie spectral sequence for Lie algebroid extensions. We show that, for wide Lie subalgebroids, the spectral sequence converges to the Lie algebroid cohomology, and that, for Lie subalgebroids over proper submanifolds, the spectral sequence converges to the formal Lie algebroid cohomology. We discuss applications and recover several constructions in Poisson geometry in which this spectral sequence has appeared naturally in the literature.
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Polynomial degeneration and the Poisson geometry of truncated polynomials
Symplectic forms on hypersurface algebroids yield generically symplectic Poisson structures whose variation along the degeneracy locus is controlled by the obstruction to lifting truncated-polynomial representations.
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