Graded Manifolds and Drinfeld Doubles for Lie Bialgebroids

By T. Voronov. 2001

170 citations recorded by the source corpus.

We define graded manifolds as a version of supermanifolds endowed with an extra Z-grading in the structure sheaf, called weight (not linked with parity). Examples are ordinary supermanifolds, vector bundles, double vector bundles (in particular, iterated constructions like TTM), etc. I give a construction of doubles for graded QS- and graded QP-manifolds (graded manifolds endowed with a homological vector field and a Schouten/Poisson bracket). Relation is explained with Drinfeld's Lie bialgebras and their doubles. Graded QS-manifolds can be considered, roughly, as "generalized Lie bialgebroids". The double for them is closely related with the analog of Drinfeld's double for Lie bialgebroids recently suggested by Roytenberg. Lie bialgebroids as a generalization of Lie bialgebras, over some base manifold, were defined by Mackenzie and P. Xu. Graded QP-manifolds give an odd version for all this, in particular, they contain "odd analogs" for Lie bialgebras, Manin triples, and Drinfeld's double.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4a4b7516f2694fb8a902dfca3940fccd80a3b8d4

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Graded bundles and homogeneity structures (2011, methodology): “We use the term ‘graded bundle’ rather than ‘graded manifold’ to distinguish our approach from those based on various different concepts of graded manifolds, usually associated with a supermanifold structure (see the discussion in [14]).”
Supergroupoids, double structures, and equivariant cohomology (2006, methodology): “We point out, however, that for Voronov’s graded manifolds, weight is allowed to be independent of parity, 8 making them somewhat more versatile than ours.”
Higher derived brackets and homotopy algebras (2003, methodology): “nonical Poisson bracket. (For the author, these expressions were a starting point in the discovery of derived brackets.) Derived brackets have been also used for describing Lie algebroids (see, e.g., [35]) and Courant algebroids [30, 28]. Higher brackets do not appear in these classical examples bec”
Derived Brackets (2003, methodology): “5) of [26] for the case of ordinary manifolds, and [56] for supermanifolds.”
Quasi, twisted, and all that... in Poisson geometry and Lie algebroid theory (2003, background): “A Lie algebroid bracket [ , ]A on a vector bundle A over a manifold M is defined, together with an anchor ρA : A → TM , by a function μ on the supermanifold T ∗ΠA ([47] [48] [55] [56]).”
Q-algebroids and their cohomology (2007, background): “The notion of basic subcomplex extends to the double complex of a double Lie algebroid [37] with generalized connection [9].”
On the structure of graded symplectic supermanifolds and Courant algebroids (2002, methodology): “Atthispointweshould rem ark thatinstead ofN-m anifolds we could consider generalgraded m anifolds [17];allour constructions extend to thiscaseatthesm allextra costofhaving to considertheeven and odd sym plectic form s separately,as the degree is no longer tied to parity.”
AKSZ–BV Formalism and Courant Algebroid-Induced Topological Field Theories (2006, unknown): “But in general these two gradings are independent from one another [12].”