82 research outputs found
The colored Jones function is q-holonomic
A function of several variables is called holonomic if, roughly speaking, it
is determined from finitely many of its values via finitely many linear
recursion relations with polynomial coefficients. Zeilberger was the first to
notice that the abstract notion of holonomicity can be applied to verify, in a
systematic and computerized way, combinatorial identities among special
functions. Using a general state sum definition of the colored Jones function
of a link in 3-space, we prove from first principles that the colored Jones
function is a multisum of a q-proper-hypergeometric function, and thus it is
q-holonomic. We demonstrate our results by computer calculations.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol9/paper29.abs.htm
Algebras of acyclic cluster type: tree type and type
In this paper, we study algebras of global dimension at most 2 whose
generalized cluster category is equivalent to the cluster category of an
acyclic quiver which is either a tree or of type . We are
particularly interested in their derived equivalence classification. We prove
that each algebra which is cluster equivalent to a tree quiver is derived
equivalent to the path algebra of this tree. Then we describe explicitly the
algebras of cluster type \A_n for each possible orientation of \A_n. We
give an explicit way to read off in which derived equivalence class such an
algebra lies, and describe the Auslander-Reiten quiver of its derived category.
Together, these results in particular provide a complete classification of
algebras which are cluster equivalent to tame acyclic quivers.Comment: v2: 37 pages. Title is changed. A mistake in the previous version is
now corrected (see Remark 3.14). Other changes making the paper coherent with
the version 2 of 1003.491
On a classification of irreducible admissible modulo representations of a -adic split reductive group
We give a classification of irreducible admissible modulo representations
of a split -adic reductive group in terms of supersingular representations.
This is a generalization of a theorem of Herzig.Comment: 25 page
One-dimensional Chern-Simons theory and the genus
We construct a Chern-Simons gauge theory for dg Lie and L-infinity algebras
on any one-dimensional manifold and quantize this theory using the
Batalin-Vilkovisky formalism and Costello's renormalization techniques. Koszul
duality and derived geometry allow us to encode topological quantum mechanics,
a nonlinear sigma model of maps from a 1-manifold into a cotangent bundle T*X,
as such a Chern-Simons theory. Our main result is that the partition function
of this theory is naturally identified with the A-genus of X. From the
perspective of derived geometry, our quantization construct a volume form on
the derived loop space which can be identified with the A-class.Comment: 61 pages, figures, final versio
Minimality and mutation-equivalence of polygons
We introduce a concept of minimality for Fano polygons. We show that, up to mutation, there are only finitely many Fano polygons with given singularity content, and give an algorithm to determine representatives for all mutation-equivalence classes of such polygons. This is a key step in a program to classify orbifold del Pezzo surfaces using mirror symmetry. As an application, we classify all Fano polygons such that the corresponding toric surface is qG-deformation-equivalent to either (i) a smooth surface; or (ii) a surface with only singularities of type 1/3(1,1)
Generalized Weyl algebras: category O and graded Morita equivalence
We study the structural and homological properties of graded Artinian modules
over generalized Weyl algebras (GWAs), and this leads to a decomposition result
for the category of graded Artinian modules. Then we define and examine a
category of graded modules analogous to the BGG category O. We discover a
condition on the data defining the GWA that ensures O has a system of
projective generators. Under this condition, O has nice
representation-theoretic properties. There is also a decomposition result for
O. Next, we give a necessary condition for there to be a strongly graded Morita
equivalence between two GWAs. We define a new algebra related to GWAs, and use
it to produce some strongly graded Morita equivalences. Finally, we give a
complete answer to the strongly graded Morita problem for classical GWAs.Comment: 19 page
Quotient closed subcategories of quiver representations
Let Q be a finite quiver without oriented cycles, and let k be an
algebraically closed field. The main result in this paper is that there is a
natural bijection between the elements in the associated Coxeter group W_Q and
the cofinite additive quotient-closed subcategories of the category of finite
dimensional right modules over kQ. We prove this correspondence by linking
these subcategories to certain ideals in the preprojective algebra associated
to Q, which are also indexed by elements of W_Q.Comment: 35 pages; v2: added a section showing how the Le-diagram condition
arises naturally from our viewpoint; v3: treat the case of hereditary
algebras over a finite fiel
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 analogue of the BGG resolution for locally analytic principal series
Let G be a connected reductive quasisplit algebraic group over a field L
which is a finite extension of the p-adic numbers. We construct an exact
sequence modelled on (the dual of) the BGG resolution involving locally
analytic principal series representations for G(L). This leads to an exact
sequence involving spaces of overconvergent p-adic automorphic forms for
certain groups compact modulo centre at infinity.Comment: 36 pages; corrected proof of Theorem 26; extended results to locally
analytic principal series for G(L); cut unnecessary expository materia
Algorithms for integrals of holonomic functions over domains defined by polynomial inequalities
We present an algorithm for computing a holonomic system for a definite
integral of a holonomic function over a domain defined by polynomial
inequalities. If the integrand satisfies a holonomic difference-differential
system including parameters, then a holonomic difference-differential system
for the integral can also be computed. In the algorithm, holonomic
distributions (generalized functions in the sense of L. Schwartz) are
inevitably involved even if the integrand is a usual function.Comment: corrected typos; Sections 5 and 6 were slightly revised with results
unchange
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