6,059 research outputs found
Categorification of Hopf algebras of rooted trees
We exhibit a monoidal structure on the category of finite sets indexed by
P-trees for a finitary polynomial endofunctor P. This structure categorifies
the monoid scheme (over Spec N) whose semiring of functions is (a P-version of)
the Connes--Kreimer bialgebra H of rooted trees (a Hopf algebra after base
change to Z and collapsing H_0). The monoidal structure is itself given by a
polynomial functor, represented by three easily described set maps; we show
that these maps are the same as those occurring in the polynomial
representation of the free monad on P.Comment: 29 pages. Does not compile with pdflatex due to dependency on the
texdraw package. v2: expository improvements, following suggestions from the
referees; final version to appear in Centr. Eur. J. Mat
Equivariant Intersection Cohomology of Toric Varieties
We investigate the equivariant intersection cohomology of a toric variety.
Considering the defining fan of the variety as a finite topological space with
the subfans being the open sets (that corresponds to the "toric" topology given
by the invariant open subsets), equivariant intersection cohomology provides a
sheaf (of graded modules over a sheaf of graded rings) on that "fan space". We
prove that this sheaf is a "minimal extension sheaf", i.e., that it satisfies
three relatively simple axioms which are known to characterize such a sheaf up
to isomorphism. In the verification of the second of these axioms, a key role
is played by "equivariantly formal" toric varieties, where equivariant and
"usual" (non-equivariant) intersection cohomology determine each other by
Kunneth type formulae. Minimal extension sheaves can be constructed in a purely
formal way and thus also exist for non-rational fans. As a consequence, we can
extend the notion of an equivariantly formal fan even to this general setup. In
this way, it will be possible to introduce "virtual" intersection cohomology
for equivariantly formal non-rational fans.Comment: 31 pages, AMS-Latex (all "private" macros included), to be published
in "Algebraic Geometry - Hirzebruch 70" (Proceedings of the conference at the
Banach Centre, Warszawa, May 1998), Contemporary Mathematics, AM
Sums over Graphs and Integration over Discrete Groupoids
We show that sums over graphs such as appear in the theory of Feynman
diagrams can be seen as integrals over discrete groupoids. From this point of
view, basic combinatorial formulas of the theory of Feynman diagrams can be
interpreted as pull-back or push-forward formulas for integrals over suitable
groupoids.Comment: 27 pages, 4 eps figures; LaTeX2e; uses Xy-Pic. Some ambiguities
fixed, and several proofs simplifie
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