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arXiv math/0011194math.QAhep-thmath.AGmath.OA
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非交换流形、瞬子代数与等谱形变

Noncommutative Manifolds the Instanton Algebra and Isospectral Deformations

Alain Connes, Giovanni Landi

备注：
We introduce the notion of admissible morphism and use it to clarify a confusing point. Definite version to appear in CMP. 20 pages, latex
发表信息：
Commun.Math.Phys.221:141-159,2001
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摘要：

我们给出了新的例子，非交换流形是不太标准的NC-环面或Moyal变形的$\Rb^n$。它们自然产生于非交换微分拓扑的基本考虑，并具有非平凡的全局特征。新的例子包括瞬子代数和NC-4-球$S^4_θ$。NC-球面上函数的非交换代数$\Ac=C^{\ify} (S^{4}_θ)$是幂等$e \in M_4 (\Ac), e^2 = e, e = e^*$的$\Ac$的循环同调中陈特征标的消失$ {\rm ch}_j (e) = 0, j < 2 $的解。由这个方程定义的泛非交换空间是由非常非平凡的三次关系定义的非交换格拉斯曼空间。这个空间${\rm Gr}$包含悬浮液的NC-3-球密切相关的量子群变形${\rm SU}_q (2)$的${\rm SU} (2)$，但不寻常的值（复数值的模量1）的参数$q$的$q$ -类似物，$q=\exp (2πi \t)$。然后，我们构造的非交换几何的$S_{\t}^4$所给出的谱三元组$(\Ac, \Hc, D)$和检查所有公理的非交换流形。非对易4-球面$S_{\t}^4$上的狄拉克算子$D$给出了定义“体积形式”$ < (e - {1/2}) [D,e]^4 > = \g_5$的基本四次方程的解，其中$<$是$4 \ts 4$矩阵在交换子上的投影。最后，我们表明，任何紧凑的黎曼自旋流形的等距群的秩$r \geq 2$承认非交换几何的等谱变形。

英文摘要：

We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of $\Rb^n$. They arise naturally from basic considerations of noncommutative differential topology and have non-trivial global features. The new examples include the instanton algebra and the NC-4-spheres $S^4_θ$. The noncommutative algebras $\Ac=C^{\ify} (S^{4}_θ)$ of functions on NC-spheres are solutions to the vanishing, $ {\rm ch}_j (e) = 0, j < 2 $, of the Chern character in the cyclic homology of $\Ac$ of an idempotent $e \in M_4 (\Ac), e^2 = e, e = e^*$. The universal noncommutative space defined by this equation is a noncommutative Grassmanian defined by very non trivial cubic relations. This space ${\rm Gr}$ contains the suspension of a NC-3-sphere intimately related to quantum group deformations ${\rm SU}_q (2)$ of ${\rm SU} (2)$ but for unusual values (complex values of modulus one) of the parameter $q$ of $q$-analogues, $q=\exp (2πi \t)$. We then construct the noncommutative geometry of $S_{\t}^4$ as given by a spectral triple $(\Ac, \Hc, D)$ and check all axioms of noncommutative manifolds. The Dirac operator $D$ on the noncommutative 4-spheres $S_{\t}^4$ gives a solution to the basic quartic equation defining the `volume form' $ < (e - {1/2}) [D,e]^4 > = \g_5$, where $<$ is the projection on the commutant of $4 \ts 4$ matrices. Finally, we show that any compact Riemannian spin manifold whose isometry group has rank $r \geq 2$ admits isospectral deformations to noncommutative geometries.

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