153 research outputs found
On a general approach to the formal cohomology of quadratic Poisson structures
We propose a general approach to the formal Poisson cohomology of -matrix
induced quadratic structures, we apply this device to compute the cohomology of
structure 2 of the Dufour-Haraki classification, and provide complete results
also for the cohomology of structure 7.Comment: 19 page
A tale of three homotopies
For a Koszul operad , there are several existing approaches to
the notion of a homotopy between homotopy morphisms of homotopy
-algebras. Some of those approaches are known to give rise to the
same notions. We exhibit the missing links between those notions, thus putting
them all into the same framework. The main nontrivial ingredient in
establishing this relationship is the homotopy transfer theorem for homotopy
cooperads due to Drummond-Cole and Vallette.Comment: 22 pages, final versio
Dequantized Differential Operators between Tensor Densities as Modules over the Lie Algebra of Contact Vector Fields
In recent years, algebras and modules of differential operators have been
extensively studied. Equivariant quantization and dequantization establish a
tight link between invariant operators connecting modules of differential
operators on tensor densities, and module morphisms that connect the
corresponding dequantized spaces. In this paper, we investigate dequantized
differential operators as modules over a Lie subalgebra of vector fields that
preserve an additional structure. More precisely, we take an interest in
invariant operators between dequantized spaces, viewed as modules over the Lie
subalgebra of infinitesimal contact or projective contact transformations. The
principal symbols of these invariant operators are invariant tensor fields. We
first provide full description of the algebras of such affine-contact- and
contact-invariant tensor fields. These characterizations allow showing that the
algebra of projective-contact-invariant operators between dequantized spaces
implemented by the same density weight, is generated by the vertical cotangent
lift of the contact form and a generalized contact Hamiltonian. As an
application, we prove a second key-result, which asserts that the Casimir
operator of the Lie algebra of infinitesimal projective contact
transformations, is diagonal. Eventually, this upshot entails that invariant
operators between spaces induced by different density weights, are made up by a
small number of building bricks that force the parameters of the source and
target spaces to verify Diophantine-type equations.Comment: 22 page
Lie algebraic characterization of manifolds
Results on characterization of manifolds in terms of certain Lie algebras
growing on them, especially Lie algebras of differential operators, are
reviewed and extended. In particular, we prove that a smooth (real-analytic,
Stein) manifold is characterized by the corresponding Lie algebra of linear
differential operators, i.e. isomorphisms of such Lie algebras are induced by
the appropriate class of diffeomorphisms of the underlaying manifolds.Comment: 15 page
Higher categorified algebras versus bounded homotopy algebras
We define Lie 3-algebras and prove that these are in 1-to-1 correspondence
with the 3-term Lie infinity algebras whose bilinear and trilinear maps vanish
in degree (1,1) and in total degree 1, respectively. Further, we give an answer
to a question of [Roy07] pertaining to the use of the nerve and normalization
functors in the study of the relationship between categorified algebras and
truncated sh algebras.Comment: 21 pages, 1 figur
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