150,920 research outputs found
Torsion theories induced from commutative subalgebras
We begin a study of torsion theories for representations of an important
class of associative algebras over a field which includes all finite W-algebras
of type A, in particular the universal enveloping algebra of gl(n) (or sl(n))
for all n. If U is such and algebra which contains a finitely generated
commutative subalgebra A, then we show that any A-torsion theory defined by the
coheight of prime ideals is liftable to U. Moreover, for any simple U-module M,
all associated prime ideals of M in Spec A have the same coheight.
Hence,thecoheight of the associated prime ideals of A is an invariant of a
given simple U-module. This implies a stratification of the category of
-modules controlled by the coheight of associated prime ideals of A. Our
approach can be viewed as a generalization of the classical paper by R.Block,
it allows in particular to study representations of gl(n) beyond the classical
category of weight or generalized weight modules
How local is the information in MPS/PEPS tensor networks?
Two dimensional tensor networks such as projected entangled pairs states
(PEPS) are generally hard to contract. This is arguably the main reason why
variational tensor network methods in 2D are still not as successful as in 1D.
However, this is not necessarily the case if the tensor network represents a
gapped ground state of a local Hamiltonian; such states are subject to many
constraints and contain much more structure. In this paper we introduce a new
approach for approximating the expectation value of a local observable in
ground states of local Hamiltonians that are represented as PEPS
tensor-networks. Instead of contracting the full tensor-network, we try to
estimate the expectation value using only a local patch of the tensor-network
around the observable. Surprisingly, we demonstrate that this is often easier
to do when the system is frustrated. In such case, the spanning vectors of the
local patch are subject to non-trivial constraints that can be utilized via a
semi-definite program to calculate rigorous lower- and upper-bounds on the
expectation value. We test our approach in 1D systems, where we show how the
expectation value can be calculated up to at least 3 or 4 digits of precision,
even when the patch radius is smaller than the correlation length.Comment: 11 pages, 5 figures, RevTeX4.1. Comments are welcome. (v2) Minor
corrections and slightly modified intro. Matches the published versio
Goal-Driven Query Answering for Existential Rules with Equality
Inspired by the magic sets for Datalog, we present a novel goal-driven
approach for answering queries over terminating existential rules with equality
(aka TGDs and EGDs). Our technique improves the performance of query answering
by pruning the consequences that are not relevant for the query. This is
challenging in our setting because equalities can potentially affect all
predicates in a dataset. We address this problem by combining the existing
singularization technique with two new ingredients: an algorithm for
identifying the rules relevant to a query and a new magic sets algorithm. We
show empirically that our technique can significantly improve the performance
of query answering, and that it can mean the difference between answering a
query in a few seconds or not being able to process the query at all
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