19 research outputs found
Invariance of the BFV-complex
The BFV-formalism was introduced to handle classical systems, equipped with
symmetries. It associates a differential graded Poisson algebra to any
coisotropic submanifold of a Poisson manifold . However the
assignment (coisotropic submanifold) (differential graded Poisson
algebra) is not canonical, since in the construction several choices have to be
made. One has to fix: 1. an embedding of the normal bundle of into
, 2. a connection on and 3. a special element . We
show that different choices of the connection and -- but with the
tubular neighbourhood fixed -- lead to isomorphic differential graded Poisson
algebras. If the tubular neighbourhood is changed too, invariance can be
restored at the level of germs.Comment: 21 pages; improved version, to appear in Pacific J. Mat
Eulerian idempotent, pre-Lie logarithm and combinatorics of trees
The aim of this paper is to bring together the three objects in the title.
Recall that, given a Lie algebra , the Eulerian idempotent is a
canonical projection from the enveloping algebra to
. The Baker-Campbell-Hausdorff product and the Magnus expansion
can both be expressed in terms of the Eulerian idempotent, which makes it
interesting to establish explicit formulas for the latter. We show how to
reduce the computation of the Eulerian idempotent to the computation of a
logarithm in a certain pre-Lie algebra of planar, binary, rooted trees. The
problem of finding formulas for the pre-Lie logarithm, which is interesting in
its own right -- being related to operad theory, numerical analysis and
renormalization -- is addressed using techniques inspired by umbral calculus.
As a consequence of our analysis, we find formulas both for the Eulerian
idempotent and the pre-Lie logarithm in terms of the combinatorics of trees.Comment: Preliminary version. Comments are welcome
How to discretize the differential forms on the interval
We provide explicit quasi-isomorphisms between the following three algebraic
structures associated to the unit interval: i) the commutative dg algebra of
differential forms, ii) the non-commutative dg algebra of simplicial cochains
and iii) the Whitney forms, equipped with a homotopy commutative and homotopy
associative, i.e. , algebra structure. Our main interest lies in a
natural `discretization' quasi-isomorphism from
differential forms to Whitney forms. We establish a uniqueness result that
implies that coincides with the morphism from homotopy transfer, and
obtain several explicit formulas for , all of which are related to the
Magnus expansion. In particular, we recover combinatorial formulas for the
Magnus expansion due to Mielnik and Pleba\'nski.Comment: 29 pages, extended abstract, typos fixe
Higher holonomies: comparing two constructions
We compare two different constructions of higher dimensional parallel
transport. On the one hand, there is the two dimensional parallel transport
associated to 2-connections on 2-bundles studied by Baez-Schreiber, Faria
Martins-Picken and Schreiber-Waldorf. On the other hand, there are the higher
holonomies associated to flat superconnections as studied by Igusa, Block-Smith
and Arias Abad-Schaetz. We first explain how by truncating the latter
construction one obtains examples of the former. Then we prove that the
2-dimensional holonomies provided by the two approaches coincide.Comment: comments are welcome
Introduction to supergeometry
These notes are based on a series of lectures given by the first author at
the school of `Poisson 2010', held at IMPA, Rio de Janeiro. They contain an
exposition of the theory of super- and graded manifolds, cohomological vector
fields, graded symplectic structures, reduction and the AKSZ-formalism.Comment: Lecture notes of a course held at the school Poisson 2010 at IMPA,
July 2010; 21 pages; references improve
A survey on stability and rigidity results for Lie algebras
We give simple and unified proofs of the known stability and rigidity results
for Lie algebras, Lie subalgebras and Lie algebra homomorphisms. Moreover, we
investigate when a Lie algebra homomorphism is stable under all automorphisms
of the codomain (including outer automorphisms).Comment: 20 page