386 research outputs found
A Poincar\'e-Birkhoff-Witt Theorem for profinite pronilpotent Lie algebras
We prove a version of the Poincar\'e-Birkhoff-Witt Theorem for profinite
pronilpotent Lie algebras in which their symmetric and universal enveloping
algebras are replaced with appropriate formal analogues and discuss some
immediate corollaries of this result.Comment: 6 page
A nondiagrammatic description of the Connes-Kreimer Hopf algebra
We demonstrate that the fundamental algebraic structure underlying the
Connes-Kreimer Hopf algebra -- the insertion pre-Lie structure on graphs --
corresponds directly to the canonical pre-Lie structure of polynomial vector
fields. Using this fact, we construct a Hopf algebra built from tensors that is
isomorphic to a version of the Connes-Kreimer Hopf algebra that first appeared
in the perturbative renormalization of quantum field theories.Comment: 25 pages, 6 figure
On the classification of Moore algebras and their deformations
In this paper we will study deformations of A-infinity algebras. We will also
answer questions relating to Moore algebras which are one of the simplest
nontrivial examples of an A-infinity algebra. We will compute the Hochschild
cohomology of odd Moore algebras and classify them up to a unital weak
equivalence. We will construct miniversal deformations of particular Moore
algebras and relate them to the universal odd and even Moore algebras. Finally
we will conclude with an investigation of formal one-parameter deformations of
an A-infinity algebra.Comment: 16 page
Noncommutative geometry and compactifications of the moduli space of curves
In this paper we show that the homology of a certain natural compactification
of the moduli space, introduced by Kontsevich in his study of Witten's
conjectures, can be described completely algebraically as the homology of a
certain differential graded Lie algebra. This two-parameter family is
constructed by using a Lie cobracket on the space of noncommutative 0-forms, a
structure which corresponds to pinching simple closed curves on a Riemann
surface, to deform the noncommutative symplectic geometry described by
Kontsevich in his subsequent papers.Comment: 23 pages, 10 figure
Classes on the moduli space of Riemann surfaces through a noncommutative Batalin-Vilkovisky formalism
Using the machinery of the Batalin-Vilkovisky formalism, we construct
cohomology classes on compactifications of the moduli space of Riemann surfaces
from the data of a contractible differential graded Frobenius algebra. We
describe how evaluating these cohomology classes upon a well-known construction
producing homology classes in the moduli space can be expressed in terms of the
Feynman diagram expansion of some functional integral. By computing these
integrals for specific examples, we are able to demonstrate that this
construction produces families of nontrivial classes.Comment: 27 pages, 4 figure
Symplectic -algebras and string topology operations
In this paper we establish the existence of certain structures on the
ordinary and equivariant homology of the free loop space on a manifold or, more
generally, a formal Poincar\'e duality space. These structures; namely the loop
product, the loop bracket and the string bracket, were introduced and studied
by Chas and Sullivan under the general heading `string topology'. Our method is
based on obstruction theory for -algebras and rational homotopy
theory. The resulting string topology operations are manifestly homotopy
invariant.Comment: Due to a strange glitch in the original submission a wrong TeX file
was uploaded; this version hopefully corrects this error. This paper is a
revision of the part of math.QA/0410621 which deals with string topology type
operations and can be read independently. 9 page
Classical theta functions from a quantum group perspective
In this paper we construct the quantum group, at roots of unity, of abelian
Chern-Simons theory. We then use it to model classical theta functions and the
actions of the Heisenberg and modular groups on them
Homotopy algebras and noncommutative geometry
We study cohomology theories of strongly homotopy algebras, namely and -algebras and establish the Hodge decomposition of
Hochschild and cyclic cohomology of -algebras thus generalising
previous work by Loday and Gerstenhaber-Schack. These results are then used to
show that a -algebra with an invariant inner product on its
cohomology can be uniquely extended to a symplectic -algebra (an
-generalisation of a commutative Frobenius algebra introduced by
Kontsevich). As another application, we show that the `string topology'
operations (the loop product, the loop bracket and the string bracket) are
homotopy invariant and can be defined on the homology or equivariant homology
of an arbitrary Poincare duality space
The topological quantum field theory of Riemann's theta functions
In this paper we prove the existence and uniqueness of a topological quantum
field theory that incorporates, for all Riemann surfaces, the corresponding
spaces of theta functions and the actions of the Heisenberg groups and modular
groups on them.Comment: 23 pages, 11 figure
Symplectic -algebras
In this paper we show that a strongly homotopy commutative (or -)
algebra with an invariant inner product on its cohomology can be uniquely
extended to a symplectic -algebra (an -generalisation of a
commutative Frobenius algebra introduced by Kontsevich). This result relies on
the algebraic Hodge decomposition of the cyclic Hochschild cohomology of a
\ci-algebra and does not generalize to algebras over other operads.Comment: This paper is a substantial revision of the part of math.QA/0410621
dealing with sympectic -algebras. The main addition is the
treatment of unital -structures. 27 page
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