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Group orbits and regular partitions of Poisson manifolds
Symplectic Geometry · Submitted 26 Sep 2006
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We study a large class of Poisson manifolds, derived from Manin triples, for which we construct explicit partitions into regular Poisson submanifolds by intersecting certain group orbits.
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The abstract, sentence by sentence
We study a large class of Poisson manifolds, derived from Manin triples, for which we construct explicit partitions into regular Poisson submanifolds by intersecting certain group orbits. Examples include all varieties ${\mathcal L}$ of Lagrangian subalgebras of reductive quadratic Lie algebras $\d$ with Poisson structures defined by Lagrangian splittings of $\d$. In the special case of $\g \oplus \g$, where $\g$ is a complex semi-simple Lie algebra, we explicitly compute the ranks of the Poisson structures on ${\mathcal L}$ defined by arbitrary Lagrangian splittings of ${\mathfrak g} \oplus {\mathfrak g}$. Such Lagrangian splittings have been classified by P. Delorme, and they contain the Belavin--Drinfeld splittings as special cases.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study a large class of Poisson manifolds, derived from Manin triples, for which we construct explicit partitions into regular Poisson submanifolds by intersecting certain group orbits. | contribution 1.00 | 0.95 | 0.58 | asserted 1.00 |
| s1 | Examples include all varieties ${\mathcal L}$ of Lagrangian subalgebras of reductive quadratic Lie algebras $\d$ with Poisson structures defined by Lagrangian splittings of $\d$. | contribution 0.60 | 0.45 | 0.10 | asserted 1.00 |
| s2 | In the special case of $\g \oplus \g$, where $\g$ is a complex semi-simple Lie algebra, we explicitly compute the ranks of the Poisson structures on ${\mathcal L}$ defined by arbitrary Lagrangian splittings of ${\mathfrak g} \oplus {\mathfrak g}$. | result 0.92 | 0.93 | 0.18 | asserted 1.00 |
| s3 | Such Lagrangian splittings have been classified by P. | background 1.00 | 0.05 | 0.03 | asserted 0.88 |
| s4 | Delorme, and they contain the Belavin--Drinfeld splittings as special cases. | background 1.00 | 0.05 | 0.04 | asserted 0.99 |
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Original abstract
We study a large class of Poisson manifolds, derived from Manin triples, for which we construct explicit partitions into regular Poisson submanifolds by intersecting certain group orbits. Examples include all varieties ${\mathcal L}$ of Lagrangian subalgebras of reductive quadratic Lie algebras $\d$ with Poisson structures defined by Lagrangian splittings of $\d$. In the special case of $\g \oplus \g$, where $\g$ is a complex semi-simple Lie algebra, we explicitly compute the ranks of the Poisson structures on ${\mathcal L}$ defined by arbitrary Lagrangian splittings of ${\mathfrak g} \oplus {\mathfrak g}$. Such Lagrangian splittings have been classified by P. Delorme, and they contain the Belavin--Drinfeld splittings as special cases.
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