2,288 research outputs found
Pre-Lie deformation theory
In this paper, we develop the deformation theory controlled by pre-Lie
algebras; the main tool is a new integration theory for pre-Lie algebras. The
main field of application lies in homotopy algebra structures over a Koszul
operad; in this case, we provide a homotopical description of the associated
Deligne groupoid. This permits us to give a conceptual proof, with complete
formulae, of the Homotopy Transfer Theorem by means of gauge action. We provide
a clear explanation of this latter ubiquitous result: there are two gauge
elements whose action on the original structure restrict its inputs and
respectively its output to the homotopy equivalent space. This implies that a
homotopy algebra structure transfers uniformly to a trivial structure on its
underlying homology if and only if it is gauge trivial; this is the ultimate
generalization of the -lemma.Comment: Final version. Minor corrections. To appear in the Moscow
Mathematical Journa
A constructive commutative quantum Lovasz Local Lemma, and beyond
The recently proven Quantum Lovasz Local Lemma generalises the well-known
Lovasz Local Lemma. It states that, if a collection of subspace constraints are
"weakly dependent", there necessarily exists a state satisfying all
constraints. It implies e.g. that certain instances of the kQSAT quantum
satisfiability problem are necessarily satisfiable, or that many-body systems
with "not too many" interactions are always frustration-free.
However, the QLLL only asserts existence; it says nothing about how to find
the state. Inspired by Moser's breakthrough classical results, we present a
constructive version of the QLLL in the setting of commuting constraints,
proving that a simple quantum algorithm converges efficiently to the required
state. In fact, we provide two different proofs, one using a novel quantum
coupling argument, the other a more explicit combinatorial analysis. Both
proofs are independent of the QLLL. So these results also provide independent,
constructive proofs of the commutative QLLL itself, but strengthen it
significantly by giving an efficient algorithm for finding the state whose
existence is asserted by the QLLL. We give an application of the constructive
commutative QLLL to convergence of CP maps.
We also extend these results to the non-commutative setting. However, our
proof of the general constructive QLLL relies on a conjecture which we are only
able to prove in special cases.Comment: 43 pages, 2 conjectures, no figures; unresolved gap in the proof; see
arXiv:1311.6474 or arXiv:1310.7766 for correct proofs of the symmetric cas
Exotic behaviour of infinite hypermaps
This is a survey of infinite hypermaps, and of how they can be constructed by using examples and techniques from combinatorial group theory, with particular emphasis on phenomena which have no analogues for finite hypermaps.<br/
-Colored Graphs - a Review of Sundry Properties
We review the combinatorial, topological, algebraic and metric properties
supported by -colored graphs, with a focus on those that are pertinent
to the study of tensor model theories. We show how to extract a limiting
continuum metric space from this set of graphs and detail properties of this
limit through the calculation of exponents at criticality
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