25,588 research outputs found
Ideal Basis in Constructions Defined by Directed Graphs
The present article continues the investigation of visible ideal bases in constructions defined using directed graphs. This notion is motivated by its applications for the design of classication systems. Our main theorem establishes that, for every balanced digraph and each idempotent semiring with identity element, the incidence semiring of the digraph has a convenient visible ideal basis. It also shows that the elements of the basis can always be used to generate ideals with the largest possible weight among the weights of all ideals in the incidence semiring
Classification theorems for the C*-algebras of graphs with sinks
We consider graphs E which have been obtained by adding one or more sinks to
a fixed directed graph G. We classify the C*-algebra of E up to a very strong
equivalence relation, which insists, loosely speaking, that C*(G) is kept
fixed. The main invariants are vectors W_E : G^0 -> N which describe how the
sinks are attached to G; more precisely, the invariants are the classes of the
W_E in the cokernel of the map A-I, where A is the adjacency matrix of the
graph G.Comment: 16 pages, uses XY-pi
Wheeled PROPs, graph complexes and the master equation
We introduce and study wheeled PROPs, an extension of the theory of PROPs
which can treat traces and, in particular, solutions to the master equations
which involve divergence operators. We construct a dg free wheeled PROP whose
representations are in one-to-one correspondence with formal germs of
SP-manifolds, key geometric objects in the theory of Batalin-Vilkovisky
quantization. We also construct minimal wheeled resolutions of classical
operads Com and Ass as rather non-obvious extensions of Com_infty and
Ass_infty, involving, e.g., a mysterious mixture of associahedra with
cyclohedra. Finally, we apply the above results to a computation of cohomology
of a directed version of Kontsevich's complex of ribbon graphs.Comment: LaTeX2e, 63 pages; Theorem 4.2.5 on bar-cobar construction is
strengthene
Three Hopf algebras from number theory, physics & topology, and their common background II: general categorical formulation
We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, co-operads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretations of known constructions in a large common framework which is presented step-by-step with examples throughout. In this second part of two papers, we give the general categorical formulation
Wheeled pro(p)file of Batalin-Vilkovisky formalism
Using technique of wheeled props we establish a correspondence between the
homotopy theory of unimodular Lie 1-bialgebras and the famous
Batalin-Vilkovisky formalism. Solutions of the so called quantum master
equation satisfying certain boundary conditions are proven to be in 1-1
correspondence with representations of a wheeled dg prop which, on the one
hand, is isomorphic to the cobar construction of the prop of unimodular Lie
1-bialgebras and, on the other hand, is quasi-isomorphic to the dg wheeled prop
of unimodular Poisson structures. These results allow us to apply properadic
methods for computing formulae for a homotopy transfer of a unimodular Lie
1-bialgebra structure on an arbitrary complex to the associated quantum master
function on its cohomology. It is proven that in the category of quantum BV
manifolds associated with the homotopy theory of unimodular Lie 1-bialgebras
quasi-isomorphisms are equivalence relations.
It is shown that Losev-Mnev's BF theory for unimodular Lie algebras can be
naturally extended to the case of unimodular Lie 1-bialgebras (and, eventually,
to the case of unimodular Poisson structures). Using a finite-dimensional
version of the Batalin-Vilkovisky quantization formalism it is rigorously
proven that the Feynman integrals computing the effective action of this new BF
theory describe precisely homotopy transfer formulae obtained within the
wheeled properadic approach to the quantum master equation. Quantum corrections
(which are present in our BF model to all orders of the Planck constant)
correspond precisely to what are often called "higher Massey products" in the
homological algebra.Comment: 42 pages. The journal versio
Crystal constructions in Number Theory
Weyl group multiple Dirichlet series and metaplectic Whittaker functions can
be described in terms of crystal graphs. We present crystals as parameterized
by Littelmann patterns and we give a survey of purely combinatorial
constructions of prime power coefficients of Weyl group multiple Dirichlet
series and metaplectic Whittaker functions using the language of crystal
graphs. We explore how the branching structure of crystals manifests in these
constructions, and how it allows access to some intricate objects in number
theory and related open questions using tools of algebraic combinatorics
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