关于非交换VNL环和GVNL环的注记

In this note, a counterexample is given to show that a noncommutative VNL-ring need not be an SVNL-ring, answering an open question of Chen and Tong(Glasgow Math. J., 48(1)(2006)) negatively. Moreover, some new results about VNL-rings and GVNL-rings are also given.     

Noncommutative Constrained KP Hierarchy and Multi-component Noncommutative Constrained KP Hierarchy

In this paper, the authors define the noncommutative constrained Kadomtsev-Petviashvili (KP) hierarchy and multi-component noncommutative constrained KP hierarchy. Then they give the recursion operators for the noncommutative constrained KP (NcKP) hierarchy and multi-component noncommutative constrained KP (NmcKP) hierarchy. The authors hope these studies might be useful in the study of D-brane dynamics whose noncommutative coordinates emerge from limits of the M theory and string theory.     

Noncommutative schemes

The main purpose of this work is to introduce noncommutative relative schemes and establish some of basic properties of schemes and scheme morphisms. In particular, we prove an analogue of the canonical bijection: ((X, O), Spec(A)) Hom (A, (X, O)). We define a noncommutative version of the ech cohomology of an affine cover and show that the ech cohomology can be used to compute higher direct images. This fact is applied here to compute cohomology of invertible sheaves on skew projective spaces and in [LR3] to study D-modules on quantum flag varieties.     

Isometries on Noncommutative Symmetric Spaces

Doklady Mathematics - Let $$\mathcal{M}$$ be an atomless semifinite von Neumann algebra equipped with a faithful normal semifinite trace τ (or else, an atomic von Neumann algebra with all...     

Noncommutative structures

We propose a method for constructing noncommutative analogs of objects from classical calculus, differential geometry, topology, dynamical systems, etc. The standard (commutative) objects can be obtained from noncommutative ones by natural projections (a set of canonical homomorphisms). The approach is ideologically close to the noncommutative geometry of A. Connes but differs from it in technical details.     

Traces on Noncommutative Homogeneous Spaces

We study properties of C*-algebraic deformations of homogeneous spaces G/Γ which are equivariant in the sense that they preserve the natural action of G by left translation. The center is shown to be isomorphic to C(G/G⁰_ρ) for a certain subgroup G⁰_ρ of G, and there is a 1-1 correspondence between normalised traces and probability measures on G/G⁰_ρ. This makes it possible to represent the deformed algebra as operators over L²(G/Γ). Applications to K-theory are also mentioned.     

Noncommutative unitons

By Uhlenbeck’s results, every harmonic map from the Riemann sphere S² to the unitary group U(n) decomposes into a product of so-called unitons: special maps from S² to the Grassmannians Gr _k(ℂⁿ) ⊂ U(n) satisfying certain systems of first-order differential equations. We construct a noncommutative analogue of this factorization, applicable to those solutions of the noncommutative unitary sigma model that are finite-dimensional perturbations of zero-energy solutions. In particular, we prove that the energy of each such solution is an integer multiple of 8π, give examples of solutions that are not equivalent to Grassmannian solutions, and study the realization of non-Grassmannian zero modes of the Hessian of the energy functional by directions tangent to the moduli space of solutions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 2, pp. 220–239, February, 2008.     

Noncommutative structures

We propose a method for constructing noncommutative analogs of objects from classical calculus, differential geometry, topology, dynamical systems, etc. The standard (commutative) objects can be obtained from noncommutative ones by natural projections (a set of canonical homomorphisms). The approach is ideologically close to the noncommutative geometry of A. Connes but differs from it in technical details. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 259, pp. 203–242.     

Noncommutative orlicz-hardy spaces

Let (Φ, Ψ) be a pair of complementary N-functions and HΦ(A)$H^{\Phi }\left(A\right)$ and HΨ(A)$H^{\Psi }\left(A\right)$ be the associated noncommutative Orlicz-Hardy spaces. We extend the Riesz, Szegö and inner-outer type factorization theorems of Hp(A)$H^p\left(A\right)$ to this case.     

Noncommutative ball maps

In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. These types of functions have recently been used in the study of dimension-free linear system engineering problems. In this paper we characterize NC analytic maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary; such maps we call “NC ball maps”. We find that up to normalization, an NC ball map is the direct sum of the identity map with an NC analytic map of the ball into the ball. That is, “NC ball maps” are very simple, in contrast to the classical result of D'Angelo on such analytic maps in C. Another mathematically natural class of maps carries a variant of the noncommutative distinguished boundary to the boundary, but on these our results are limited. We shall be interested in several types of noncommutative balls, conventional ones, but also balls defined by constraints called Linear Matrix Inequalities (LMI). What we do here is a small piece of the bigger puzzle of understanding how LMIs behave with respect to noncommutative change of variables.     

Noncommutative cohomologies and Mumford groups

In this note we prove the “quadratic property” of the boundary mapping in the exact sequence of cohomologies. We present calculations for the action of the Galois group on the Mumford group.     

Noncommutative topological dynamics

We study noncommutative dynamical systems associated to unimodal and bimodal maps of the interval. To these maps we associate subshifts and the correspondent AF-algebras and Cuntz–Krieger algebras. As an example we consider systems having equal topological entropy log(1 + ϕ), where ϕ is the golden number, but distinct chaotic behavior and we show how a new numerical invariant allows to distinguish that complexity. Finally, we give a statistical interpretation to the topological numerical invariants associated to bimodal maps.     

Noncommutative cyclic covers and maximal orders on surfaces

In this paper, we construct various examples of maximal orders on surfaces, including some del Pezzo orders, some ruled orders and some numerically Calabi-Yau orders. The method of construction is a noncommutative version of the cyclic covering trick. These noncommutative cyclic covers are very computable and we give a formula for their ramification data. This often allows us to determine if a maximal order, described via ramification data, can be constructed as a noncommutative cyclic cover. The construction also has applications to Brauer-Severi varieties and, in the quaternion case, we show how to obtain some Brauer-Severi varieties from G-Hilbert schemes of P¹-bundles.     

Noncommutative Hamiltonian dynamics on foliated manifolds

First, we review the notion of a Poisson structure on a noncommutative algebra due to Block, Getzler and Xu, and we introduce a notion of a Hamiltonian vector field on a noncommutative Poisson algebra. Then we describe a Poisson structure on a noncommutative algebra associated with a transversely symplectic foliation and construct a class of Hamiltonian vector fields associated with this Poisson structure.     

The McCoy Condition on Noncommutative Rings

We introduce the multiplication algebra of a Bernstein algebra, establish its Peirce decomposition relative to an idempotent of A and state some basic properties of this algebra of endomorphtsms     

Noncommutative local algebra

The main purpose of this work is to introduce the first notions of noncommutative algebraic geometry — the spectrum of an abelian category, localizations at points of the spectrum, canonical topologies, supports, associated points etc. — and to study their basic properties.     

Noncommutative Artin motives

In this article, we introduce the category of noncommutative Artin motives as well as the category of noncommutative mixed Artin motives. In the pure world, we start by proving that the classical category ${{\mathrm{AM}}}(k)_\mathbb Q $ of Artin motives (over a base field k) can be characterized as the largest category inside Chow motives which fully embeds into noncommutative Chow motives. Making use of a refined bridge between pure motives and noncommutative pure motives, we then show that the image of this full embedding, which we call the category ${{\mathrm{NAM}}}(k)_\mathbb Q $ of noncommutative Artin motives, is invariant under the different equivalence relations and modification of the symmetry isomorphism constraints. As an application, we recover the absolute Galois group $\mathrm{Gal}(\overline{k}/k)$ from the Tannakian formalism applied to ${{\mathrm{NAM}}}(k)_\mathbb Q $ . Then, we develop the base-change formalism in the world of noncommutative pure motives. As an application, we obtain new tools for the study of motivic decompositions and Schur/Kimura finiteness. Making use of this theory of base-change, we then construct a short exact sequence relating $\mathrm{Gal}(\overline{k}/k)$ with the noncommutative motivic Galois groups of k and $\overline{k}$ . Finally, we describe a precise relationship between this short exact sequence and the one constructed by Deligne–Milne. In the mixed world, we introduce the triangulated category ${{\mathrm{NMAM}}}(k)_\mathbb Q $ of noncommutative mixed Artin motives and construct a faithful functor from the classical category ${{\mathrm{MAM}}}(k)_\mathbb Q $ of mixed Artin motives to it. When k is a finite field, this functor is an equivalence. On the other hand, when k is of characteristic zero ${{\mathrm{NMAM}}}(k)_\mathbb Q $ is much richer than ${{\mathrm{MAM}}}(k)_\mathbb Q $ since its higher Ext-groups encode all the (rationalized) higher algebraic $K$ -theory of finite étale k-schemes. In the appendix, we establish a general result about short exact sequences of Galois groups which is of independent interest. As an application, we obtain a new proof of Deligne–Milne’s short exact sequence.