3,721 research outputs found
Tensor product categorifications and the super Kazhdan-Lusztig conjecture
We give a new proof of the "super Kazhdan-Lusztig conjecture" for the Lie
super algebra as formulated originally by the
first author. We also prove for the first time that any integral block of
category O for (and also all of its parabolic
analogs) possesses a graded version which is Koszul. Our approach depends
crucially on an application of the uniqueness of tensor product
categorifications established recently by the second two authors.Comment: 58 pages; v2: relatively minor changes, a few adjustments to wording
and references; v3: final version, more minor changes, to appear in IMR
Recovering the topology of surfaces from cluster algebras
We present an effective method for recovering the topology of a bordered
oriented surface with marked points from its cluster algebra. The information
is extracted from the maximal triangulations of the surface, those that have
exchange quivers with maximal number of arrows in the mutation class. The
method gives new proofs of the automorphism and isomorphism problems for the
surface cluster algebras, as well as the uniqueness of the
Fomin-Shapiro-Thurston block decompositions of the exchange quivers of the
surface cluster algebras. The previous proofs of these results followed a
different approach based on Gu's direct proof of the last result. The method
also explains the exceptions to these results due to pathological problems with
the maximal triangulations of several surfaces.Comment: 29 pages, AMS Late
Universal Constructions for (Co)Relations: categories, monoidal categories, and props
Calculi of string diagrams are increasingly used to present the syntax and
algebraic structure of various families of circuits, including signal flow
graphs, electrical circuits and quantum processes. In many such approaches, the
semantic interpretation for diagrams is given in terms of relations or
corelations (generalised equivalence relations) of some kind. In this paper we
show how semantic categories of both relations and corelations can be
characterised as colimits of simpler categories. This modular perspective is
important as it simplifies the task of giving a complete axiomatisation for
semantic equivalence of string diagrams. Moreover, our general result unifies
various theorems that are independently found in literature and are relevant
for program semantics, quantum computation and control theory.Comment: 22 pages + 3 page appendix, extended version of arXiv:1703.0824
Nica-Toeplitz algebras associated with product systems over right LCM semigroups
We prove uniqueness of representations of Nica-Toeplitz algebras associated
to product systems of -correspondences over right LCM semigroups by
applying our previous abstract uniqueness results developed for
-precategories. Our results provide an interpretation of conditions
identified in work of Fowler and Fowler-Raeburn, and apply also to their
crossed product twisted by a product system, in the new context of right LCM
semigroups, as well as to a new, Doplicher-Roberts type -algebra
associated to the Nica-Toeplitz algebra. As a derived construction we develop
Nica-Toeplitz crossed products by actions with completely positive maps. This
provides a unified framework for Nica-Toeplitz semigroup crossed products by
endomorphisms and by transfer operators. We illustrate these two classes of
examples with semigroup -algebras of right and left semidirect products.Comment: Title changed from "Nica-Toeplitz algebras associated with right
tensor C*-precategories over right LCM semigroups: part II examples". The
manuscript accepted in J. Math. Anal. App
Basic deformation theory of smooth formal schemes
We provide the main results of a deformation theory of smooth formal schemes.
First we deal with the case of global lifting of smooth morphisms. We prove
that the obstruction to the existence of a global lifting lies in a Ext^1
group. Then we study uniqueness and existence of lifting of smooth formal
schemes. The set of isomorphism classes of smooth liftings is classified by a
Ext^1 group and there exists an obstruction in a Ext^2 group whose vanishing
characterizes the existence of smooth liftings.Comment: 14 page
Compactification of Drinfeld modular varieties and Drinfeld Modular Forms of Arbitrary Rank
We give an abstract characterization of the Satake compactification of a
general Drinfeld modular variety. We prove that it exists and is unique up to
unique isomorphism, though we do not give an explicit stratification by
Drinfeld modular varieties of smaller rank which is also expected. We construct
a natural ample invertible sheaf on it, such that the global sections of its
-th power form the space of (algebraic) Drinfeld modular forms of weight
. We show how the Satake compactification and modular forms behave under all
natural morphisms between Drinfeld modular varieties; in particular we define
Hecke operators. We give explicit results in some special cases
The Uniqueness Theorem for Entanglement Measures
We explore and develop the mathematics of the theory of entanglement
measures. After a careful review and analysis of definitions, of preliminary
results, and of connections between conditions on entanglement measures, we
prove a sharpened version of a uniqueness theorem which gives necessary and
sufficient conditions for an entanglement measure to coincide with the reduced
von Neumann entropy on pure states. We also prove several versions of a theorem
on extreme entanglement measures in the case of mixed states. We analyse
properties of the asymptotic regularization of entanglement measures proving,
for example, convexity for the entanglement cost and for the regularized
relative entropy of entanglement.Comment: 22 pages, LaTeX, version accepted by J. Math. Phy
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