Related papers: Integral Calculus on Quantum Exterior Algebras

We set up a left ring of fractions over a certain ring of boundary problems for linear ordinary differential equations. The fraction ring acts naturally on a new module of generalized functions. The latter includes an isomorphic copy of the…

For any congruence subgroup $\Gamma$, we study the vertex operator algebra $\Omega^{ch}(\mathbb H,\Gamma)$ constructed from the $\Gamma$-invariant global sections of the chiral de Rham complex on the upper half plane, which are holomorphic…

In 1960-62, E. K\"ahler developed what looks as a generalization of the exterior calculus, which he based on Clifford rather than exterior algebra. The role of\ the exterior derivative, $du$, was taken by the more comprehensive derivative…

The quantum cohomology algebra of a projective manifold X is the cohomology H(X,Q) endowed with a different algebra structure, which takes into account the geometry of rational curves in X. We show that this algebra takes a remarkably…

We show how one can construct a differential calculus over an algebra where position variables x and momentum variables p have be defined. As the simplest example we consider the one-dimensional q-deformed Heisenberg algebra. This algebra…

We build a differential calculus for subalgebras of the Moyal algebra on R^4 starting from a redundant differential calculus on the Moyal algebra, which is suitable for reduction. In some cases we find a frame of 1-forms which allows to…

This paper defines and examines the basic properties of noncommutative analogues of almost complex structures, integrable almost complex structures, holomorphic curvature, cohomology, and holomorphic sheaves. The starting point is a…

We develop a family of finite element spaces of differential forms defined on cubical meshes in any number of dimensions. The family contains elements of all polynomial degrees and all form degrees. In two dimensions, these include the…

A new framework for noncommutative complex geometry on quantum homogeneous spaces is introduced. The main ingredients used are covariant differential calculi and Takeuchi's categorical equivalence for faithfully flat quantum homogeneous…

Quantum Chern-Simons invariants of differentiable manifolds are analyzed from the point of view of homological algebra. Given a manifold M and a Lie (or, more generally, an L-infinity) algebra g, the vector space H^*(M) \otimes g has the…

We study the cohomology ring of the complement $\mathcal{M}(\mathcal{A})$ of a manifold arrangement $\mathcal{A}$ in a smooth manifold $M$ without boundary. We first give the concept of monoidal cosheaf on a locally geometric poset…

We introduce a new topological invariant of complex line arrangements in the complex projective plane, derived from the interaction between their complement and the boundary of a regular neighbourhood. The motivation is to identify Zariski…

Let $\cl{M}$ be a Hilbert module of holomorphic functions over a natural function algebra $\mathcal{A}(\Omega)$, where $\Omega \subseteq \bb{C}^m$ is a bounded domain. Let $\cl{M}_0\subseteq \cl{M}$ be the submodule of functions vanishing…

We introduce the notion of being cohomologically complete for objects of the derived category of sheaves of $Z[\hbar]$-modules on a topological space. Then we consider a $Z[\hbar]$-algebra satisfying some suitable conditions and prove…

Many coupled problems in engineering and science can be described by elliptic partial differential equations on adjacent domains, where the coupling can be considered either as a thin equidimensional overlap between the model domains, or as…

We study real and integral structures in the space of solutions to the quantum differential equations. First we show that, under mild conditions, any real structure in orbifold quantum cohomology yields a pure and polarized tt^*-geometry…

Let $X=\C^n$. In this paper we present an algorithm that computes the de Rham cohomology groups $H^i_{dR}(U,\C)$ where $U$ is the complement of an arbitrary Zariski-closed set $Y$ in $X$. Our algorithm is a merger of the algorithm given by…

We give a systematic and self-contained account of the construction of geometrically decomposed bases and degrees of freedom in finite element exterior calculus. In particular, we elaborate upon a previously overlooked basis for one of the…

For a maximal separable subfield $K$ of a central simple algebra $A$, we provide a semiring isomorphism between $K$-$K$-bimodules $A$ and $H$-$H$ bisets of $G = \Gal(L/F)$, where $F = \operatorname{Z}(A)$, $L$ is the Galois closure of…

The Rumin algebra of a contact manifold is a contact invariant $C_\infty$-algebra of differential forms which computes the de Rham cohomology algebra. We recover this fact by giving a simple and explicit construction of the Rumin algebra…