373 research outputs found
Operads with compatible CL-shellable partition posets admit a Poincar\'e-Birkhoff-Witt basis
In 2007, Vallette built a bridge across posets and operads by proving that an
operad is Koszul if and only if the associated partition posets are
Cohen-Macaulay. Both notions of being Koszul and being Cohen-Macaulay admit
different refinements: our goal here is to link two of these refinements. We
more precisely prove that any (basic-set) operad whose associated posets admit
isomorphism-compatible CL-shellings admits a Poincar\'e-Birkhoff-Witt basis.
Furthermore, we give counter-examples to the converse
Finite groups acting on homology manifolds
In this paper we study homology manifolds T admitting the action of a finite group preserving the structure of a regular CW-complex on T. The CW-complex is parameterized by a poset and the topological properties of the manifold are translated into a combinatorial setting via the poset. We concentrate on n-manifolds which admit a fairly rigid group of automorphisms transitive on the n-cells of the complex. This allows us to make yet another translation from a combinatorial into a group theoretic setting. We close by using our machinery to construct representations on manifolds of the Monster, the largest sporadic group. Some of these manifolds are of dimension 24, and hence candidates for examples to Hirzebruch's Prize Question in [HBJ], but unfortunately closer inspection shows the A^-genus of these manifolds is 0 rather than 1, so none is a Hirzebruch manifold
The order topology for a von Neumann algebra
The order topology (resp. the sequential order topology
) on a poset is the topology that has as its closed sets
those that contain the order limits of all their order convergent nets (resp.
sequences). For a von Neumann algebra we consider the following three
posets: the self-adjoint part , the self-adjoint part of the unit ball
, and the projection lattice . We study the order topology (and
the corresponding sequential variant) on these posets, compare the order
topology to the other standard locally convex topologies on , and relate the
properties of the order topology to the underlying operator-algebraic structure
of
Connectivity Properties of Factorization Posets in Generated Groups
We consider three notions of connectivity and their interactions in partially
ordered sets coming from reduced factorizations of an element in a generated
group. While one form of connectivity essentially reflects the connectivity of
the poset diagram, the other two are a bit more involved: Hurwitz-connectivity
has its origins in algebraic geometry, and shellability in topology. We propose
a framework to study these connectivity properties in a uniform way. Our main
tool is a certain linear order of the generators that is compatible with the
chosen element.Comment: 35 pages, 17 figures. Comments are very welcome. Final versio
On Derived Equivalences of Categories of Sheaves Over Finite Posets
A finite poset X carries a natural structure of a topological space. Fix a
field k, and denote by D(X) the bounded derived category of sheaves of finite
dimensional k-vector spaces over X. Two posets X and Y are said to be derived
equivalent if D(X) and D(Y) are equivalent as triangulated categories.
We give explicit combinatorial properties of a poset which are invariant
under derived equivalence, among them are the number of points, the
Z-congruency class of the incidence matrix, and the Betti numbers.
Then we construct, for any closed subset Y of X, a strongly exceptional
collection in D(X) and use it to show an equivalence between D(X) and the
bounded derived category of a finite dimensional algebra A (depending on Y). We
give conditions on X and Y under which A becomes an incidence algebra of a
poset.
We deduce that a lexicographic sum of a collection of posets along a
bipartite graph is derived equivalent to the lexicographic sum of the same
collection along the opposite graph. This construction produces many new
derived equivalences of posets and generalizes other well known ones.
As a corollary we show that the derived equivalence class of an ordinal sum
of two posets does not depend on the order of summands. We give an example that
this is not true for three summands.Comment: 20 page
An uncountable Mittag-Leffler condition with an application to ultrametric locally convex vector spaces
Mittag-Leffler condition ensures the exactness of the inverse limit of short
exact sequences indexed on a partially ordered set admitting a
cofinal subset. We extend Mittag-Leffler condition by relatively
relaxing the countability assumption. As an application we prove an ultrametric
analogous of a result of V.P.Palamodov in relation with the acyclicity of
Frechet spaces with respect to the completion functor.Comment: 19 page
A new light on nets of C*-algebras and their representations
The present paper deals with the question of representability of nets of
C*-algebras whose underlying poset, indexing the net, is not upward directed. A
particular class of nets, called C*-net bundles, is classified in terms of
C*-dynamical systems having as group the fundamental group of the poset. Any
net of C*-algebras embeds into a unique C*-net bundle, the enveloping net
bundle, which generalizes the notion of universal C*-algebra given by
Fredenhagen to nonsimply connected posets. This allows a classification of
nets; in particular, we call injective those nets having a faithful embedding
into the enveloping net bundle. Injectivity turns out to be equivalent to the
existence of faithful representations. We further relate injectivity to a
generalized Cech cocycle of the net, and this allows us to give examples of
nets exhausting the above classification. Using the results of this paper we
shall show, in a forthcoming paper, that any conformal net over S^1 is
injective
Multifraction reduction III: The case of interval monoids
We investigate gcd-monoids, which are cancellative monoids in which any two
elements admit a left and a right gcd, and the associated reduction of
multifractions (arXiv:1606.08991 and 1606.08995), a general approach to the
word problem for the enveloping group. Here we consider the particular case of
interval monoids associated with finite posets. In this way, we construct
gcd-monoids, in which reduction of multifractions has prescribed properties not
yet known to be compatible: semi-convergence of reduction without convergence,
semi-convergence up to some level but not beyond, non-embeddability into the
enveloping group (a strong negation of semi-convergence).Comment: 23 pages ; v2 : cross-references updated ; v3 : one example added,
typos corrected; final version due to appear in Journal of Combinatorial
Algebr
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