Affine transformation crossed product like algebras and noncommutative surfaces

Classification of digital affine noncommutative geometries

It is known that connected translation invariant $n$-dimensional noncommutative differentials $d x^i$ on the algebra $k[x^1,\cdots,x^n]$ of polynomials in $n$-variables over a field $k$ are classified by commutative algebras $V$ on the vector space spanned by the coordinates. This data also applies to construct differentials on the Heisenberg algebra `spacetime' with relations $[x^μ,x^ν]=λΘ^{μν}$ where $ Θ$ is an antisymmetric matrix as well as to Lie algebras with pre-Lie algebra structures. We specialise the general theory to the field $k={\ \mathbb{F}}_2$ of two elements, in which case tran…

晶格模型中的拓扑缺陷与仿射 Temperley–Lieb 代数

Topological defects in lattice models and affine Temperley-Lieb algebra

论文以代数方法从交叉道和直接道定义临界晶格模型中的拓扑缺陷，揭示仿射 Temperley–Lieb 代数中心的融合基，并引入融合积与融合商描述含缺陷模型的表示和谱。

从非交换动力学到 $U(1)_{\star}$ 电动力学：一族具有局部化 gauge 诱导扰动的谱三元组

Affine Left--Right Dirac Data on the Moyal Plane, with an Application to $U(1)_\star$ Gauge Coupling in Noncommutative Quantum Mechanics

本文构建了非交换平面系统的谱三元组框架，通过线性Darboux规范和Stone-von Neumann定理，将光滑算子代数与有效Moyal侧Frechet *-代数关联，从而得到局部紧非单位的谱三元组，并证明有限截断谱三元组在极限下逼近最小耦合Dirac算子。

Deformation Expression for Elements of Algebras (II) --(Weyl algebra of 2m-generators)--

This is a noncommutative version of the previous work entitled "Deformation Expression for Elements of Algebras (I)." In general in a noncommutative algebra, there is no canonical way to express elements in univalent way, which is often called "ordering problem". In this note we discuss this problem in the case of the Weyl algebra of 2m-generators. By fixing an expression, we extends Weyl algebra transcendentally. We treat *-exponential functions of linear forms, and quadratic forms of crossed symbol under generic expression parameters.

Real Spectral Triples on Crossed Products

Given a spectral triple on a unital $C^{*}$-algebra $A$ and an equicontinuous action of a discrete group $G$ on $A$, a spectral triple on the reduced crossed product $C^{*}$-algebra $A\rtimes_r G$ was constructed by Hawkins, Skalski, White and Zacharias in [On spectral triples on crossed products arising from equicontinuous actions, Math. Scand. 113(2) (2013) 262-291], extending the construction by Belissard, Marcolli and Reihani in [Dynamical systems on spectral metric spaces, preprint (2010), arXiv:1008.4617], by using the Kasparov product to make an ansatz for the Dirac operator. Supposing …

螺旋膜、仿射 qq-特征标与椭圆可积系统

Spiralling branes, affine qq-characters and elliptic integrable systems

本文将螺旋膜技术推广到多体可积系统，利用量子环面代数交织子重新描述三角 RS 系统，并研究其椭圆形变与 Shiraishi 函数的关系，证明仿射 qq-特征标的完全非对易 Jacobi 恒等式。

Noncommutative Schur polynomials and the crystal limit of the U_q sl(2)-vertex model

Starting from the Verma module of U_q sl(2) we consider the evaluation module for affine U_q sl(2) and discuss its crystal limit (q=0). There exists an associated integrable statistical mechanics model on a square lattice defined in terms of vertex configurations. Its transfer matrix is the generating function for noncommutative complete symmetric polynomials in the generators of the affine plactic algebra, an extension of the finite plactic algebra first discussed by Lascoux and Schützenberger. The corresponding noncommutative elementary symmetric polynomials were recently shown to be generat…

The Picard Group of a Noncommutative Algebraic Torus

We compute the Picard group $ Pic(A_q) $ of the noncommutative algebraic 2-torus $A_q$, describe its action on the space $ R(A_q) $ of isomorphism classes of rk 1 projective modules and classify the algebras Morita equivalent to $ A_q $. Our computations are based on a quantum version of the Calogero-Moser correspondence relating projective $A_q$-modules to irreducible representations of the double affine Hecke algebras (DAHA) $ H_{t, q^{-1/2}}(S_n) $ at $ t = 1 $. We show that, under this correspondence, the action of $ Pic(A_q) $ on $ R(A_q) $ agrees with the action of $ SL_2(Z) $ on $ H_{t,…

The calculus of multivectors on noncommutative jet spaces

The Leibniz rule for derivations is invariant under cyclic permutations of co-multiples within the arguments of derivations. We explore the implications of this principle: in effect, we construct a class of noncommutative bundles in which the sheaves of algebras of walks along a tesselated affine manifold form the base, whereas the fibres are free associative algebras or, at a later stage, such algebras quotients over the linear relation of equivalence under cyclic shifts. The calculus of variations is developed on the infinite jet spaces over such noncommutative bundles. In the frames of such…

Clifford and Riemann-Finsler Structures in Geometric Mechanics and Gravity

The book contains a collection of works on Riemann-Cartan and metric-affine manifolds provided with nonlinear connection structure and on generalized Finsler-Lagrange and Cartan-Hamilton geometries and Clifford structures modelled on such manifolds. The choice of material presented has evolved from various applications in modern gravity and geometric mechanics and certain generalizations to noncommutative Riemann-Finsler geometry. The authors develop and use the method of anholonomic frames with associated nonlinear connection structure and apply it to a number of concrete problems: constructi…

Exotic smooth R^4, noncommutative algebras and quantization

The paper shows deep connections between exotic smoothings of small R^4, noncommutative algebras of foliations and quantization. At first, based on the close relation of foliations and noncommutative C*-algebras we show that cyclic cohomology invariants characterize some small exotic R^4. Certain exotic smooth R^4's define a generalized embedding into a space which is K-theoretic equivalent to a noncommutative Banach algebra. Furthermore, we show that a factor III von Neumann algebra is naturally related with nonstandard smoothing of a small R^4 and conjecture that this factor is the unique hy…