We develop a sheaf theory approach to toric noncommutative geometry which allows us to formalize the concept of mapping spaces between two toric noncommutative spaces. As an application we study the `internalized' automorphism group of a toric noncommutative space and show that its Lie algebra has an elementary description in terms of braided derivations.
We introduce and study several new topological operads that should be regarded as nonsymmetric analogues of the operads of little 2-disks, framed little 2-disks, and Deligne-Mumford compactifications of moduli spaces of genus zero curves with marked points. These operads exhibit all the remarkable algebraic and geometric features that their classical analogues possess; in particular, it is possible to define a noncommutative analogue of the notion of cohomological field theory with similar Givental-type symmetries. This relies on rich geometry of the analogues of the Deligne-Mumford spaces, co…
We address a natural question in noncommutative geometry, namely the rigidity observed in many examples, whereby noncommutative spaces (or equivalently their coordinate algebras) have very few automorphisms by comparison with their commutative counterparts. In the framework of noncommutative projective geometry, we define a groupoid whose objects are noncommutative projective spaces of a given dimension and whose morphisms correspond to isomorphisms of these. This groupoid is then a natural generalization of an automorphism group. Using work of Zhang, we may translate this structure to the alg…
We consider the algebra $A$ of bounded operators on $L^2(\mathbb{R}^n)$ generated by quantizations of isometric affine canonical transformations. The algebra $A$ includes as subalgebras all noncommutative tori and toric orbifolds. We define the spectral triple $(A, H, D)$ with $H=L^2(\mathbb R^n, Λ(\mathbb R^n))$ and the Euler operator $D$, a first order differential operator of index $1$. We show that this spectral triple has simple dimension spectrum: For every operator $B$ in the algebra $Ψ(A,H,D)$ generated by the Shubin type pseudodifferential operators and the elements of $A$, the zeta f…
We study noncommutative principal bundles (Hopf-Galois extensions) in the context of coquasitriangular Hopf algebras and their monoidal category of comodule algebras. When the total space is quasi-commutative, and thus the base space subalgebra is central, we define the gauge group as the group of vertical automorphisms or equivalently as the group of equivariant algebra maps. We study Drinfeld twist (2-cocycle) deformations of Hopf-Galois extensions and show that the gauge group of the twisted extension is isomorphic to the gauge group of the initial extension. In particular, noncommutative p…
For any number $h$ such that $\hbar:=h/2π$ is irrational and any skew-symmetric, non-degenerate bilinear form $σ:\mathbb{Z}^{2g}\times \mathbb{Z}^{2g} \to \mathbb{Z}$, let be $\mathcal{A}^h_{g,σ}$ be the twisted group $*$-algebra $\mathbb{C}[\mathbb{Z}^{2g}]$ and consider the ergodic group of $*$-automorphisms of $\mathcal{A}^h_{g,σ}$ induced by the action of the symplectic group Sp$(\mathbb{Z}^{2g},σ)$. We show that the only Sp$(\mathbb{Z}^{2g},σ)$-invariant state on $\mathcal{A}^h_{g,σ}$ is the trace state $τ$.
From quantum Ore extensions to quantum tori via noncommutative UFDs
所有产生于Kac-Moody代数的量子化泛包络代数的迭代斜多项式扩张是一个非常大的公理定义的代数类的特例，称为CGL扩张。为了在代数R上构造量子团簇代数结构的初始团簇，以及对R的自同构进行分类，需要将R嵌入到量子环面T中，且嵌入具有R包含相应的量子仿射空间代数A的性质.我们显式地构造这样一个嵌入A \subseteq R \subset T的每个CGL扩展R使用的方法的非交换诺特唯一因子分解域和运行一个Gelfand-Tsetlin型程序正常，而不是中心元素。一路上，我们分类的齐次素元的所有CGL扩展，我们证明了每个CGL扩展R有一个相关的极大环面，它涵盖了自同构的R对应的所有正常的元素。对于对称CGL扩张，我们描述了量子仿射空间代数A与Cauchon的量子仿射空间代数A之间的关系.
