Graded manifolds and Drinfeld doubles for Lie bialgebroids. (English) Zbl 1042.53056

Voronov, Theodore (ed.), Quantization, Poisson brackets and beyond. London Mathematical Society regional meeting and workshop on quantization, deformations, and new homological and categorical methods in mathematical physics, Manchester, UK, July 6–13, 2001. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3201-8). Contemp. Math. 315, 131-168 (2002).
The classical Drinfeld double of a Lie bialgebra is again a Lie bialgebra, with some nice properties. Various generalizations (due to Mackenzie, Liu-Weinstein-Xu and Roytenberg) of the double construction to Lie bialgebroids all produced objects which were not Lie bialgebroids, and not even Lie algebroids. In the work of Roytenberg, it was suggested that the supermanifold \(T^{*}(\Pi E)\) with a certain natural structure should be considered as the double for a Lie bialgebroid \(E\to X\) over a base \( X \). The author generalizes Roytenberg’s construction to a newly defined class of supermanifolds (called graded manifolds), and investigates the properties of the resulting doubles.
In the paper, a graded manifold is defined as a supermanifold with an additional \(\mathbb {Z}\)-grading in the structure sheaf. This \(\mathbb {Z}\)-grading is completely independent of the \(\mathbb {Z}_{2}\)-grading of the supermanifold, and is called weight. Examples of graded manifolds are the usual supermanifolds (with the zero weight assigned to all coordinates), vector bundles (with the same weight non-zero integer assigned to all linear fiber coordinates, and zero weight assigned to all base coordinates), double vector bundles (e.g., a (co)tangent bundle of a given vector bundle), etc. On a graded manifold, all geometric objects (e.g., vector fields, differential forms, Poisson brackets, etc.) are automatically assigned weight.
In order to generalize the double construction to graded manifolds, the author shows that a Lie bialgebra structure on \(\mathfrak {g} \) is equivalent to a homological vector field (representing the bracket on \(\mathfrak {g} \) and a Schouten bracket (representing the cobracket) on \(\Pi \mathfrak {g} \) (where \(\Pi \) is the change of parity functor), with a certain compatibility condition. The double of \(\mathfrak {g} \) corresponds to a Hamiltonian homological vector field and a Schouten bracket on \(T^{*}\Pi \mathfrak {g}(\simeq \Pi (\mathfrak {g}\oplus \mathfrak {g}^{*}))\).
The author considers a graded supermanifold \( M \) endowed with a homological vector field \(\hat{Q} \) of weight \(q \) and a Schouten bracket \([\, ,\, ] \) of weight \(s \), which satisfy a compatibility condition. He shows that on the space \(DM=T^{*}M \) there is a homological vector field \( \hat{Q}_{D} \) of the same weight \(q \) such that \((DM,\widehat{Q}_{D})\) can be considered as the double of \((M,\hat{Q},[\, ,\, ]) \). This double inherits half of the original structure — the homological vector field. Using a linear connection on \(M \), it is possible to define an almost-Schouten bracket on \(T^{*}M \) as well. In the case when \(M \) comes from a Lie superalgebra, this construction recovers the standard Drinfeld double. As an example, the author considers odd Lie bialgebras and odd double.
For the entire collection see [Zbl 1007.53002].


53D17 Poisson manifolds; Poisson groupoids and algebroids
58A50 Supermanifolds and graded manifolds
17B62 Lie bialgebras; Lie coalgebras
17B63 Poisson algebras
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