Questions tagged [noncommutative-geometry]
Noncommutative geometry is a study of noncomutative algebras from geometrical point of view. The motivation of this approach is Gelfand representation theorem, which shows that every commutative C*-algebra is *-isomorphic to the space of continuous functions on some locally compact Hausdorff space.
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Constructing Cyclic Division Algebras
I'm studying the construction of cyclic division algebras but I don't see how a given example holds.
According to the literature, we start with a finite extension $L$ of a number field $K$ such that ...
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Index of Callias operator and application in physics
In his article "Axial Anomalies and Index Theorems on Open Spaces" (https://link.springer.com/article/10.1007/BF01202525) C.Callias shows how the index of the Callias-type operator on $R^{n}$...
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Why do Kontsevich and Rosenberg use algebra epimorphisms rather than surjections?
In the article Noncommutative spaces Kontsevich and Rosenberg make the following definition:
2.6. The Q-category of infinitesimal algebra epimorphisms. Let $A$ be the
category $Alg_k$ of associative ...
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Technical question about computing the left ideals of the ring $\mathbb{Z}_{2} \oplus \mathbb{Z}_{2} \oplus \mathbb{Z}_{2}$
Computing the left ideals of $\mathbb{Z}_{2} \oplus \mathbb{Z}_{2} \oplus \mathbb{Z}_{2}$. This is the same that computing the ideals of $\mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2}$, ...
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Lie algebra cohomology of formal non-commutative vector fields
Let $k$ be a field of characteristic $0$ and $A=k\langle\langle x_1,\dotsc,x_n\rangle\rangle$ be a free completed associative algebra. The space of continuous derivations $\mathrm{Der}(A)$ is ...
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Definition of Fredholm modules
I'm currently starting to learn K-homology and something bothers me about the definition of Fredholm modules. I looked in 5/6 papers and each time a different definition is given... For example in [1] ...
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Lattices and noncommutative algebras in noncommutative geometry
I am interested in the relation between lattices and noncommutative algebras in the context of noncommutative geometry. In the commutative case, a lattice is a discrete subgroup of a locally compact ...
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Vector-valued differential forms and cyclic homology
Let $F\rightarrow E \rightarrow M$, where $E$ - smooth flat bundle, $M$ - smooth compact manifold, $F$ - (commutative) algebra over $\mathbb{C}$. Is it true that local cyclic homology of $\Gamma ^ \...
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Question Regarding the Eigenvalues of Matrix Commutators
Let us consider a commutator of two diagonizable $n\times n$, i.e. square matrices, $A$ and $B$. The eigenvalues of $A$ are $\lambda^1,\lambda^2,...,\lambda^{n}$ The eigen values of $B$ are $\mu^1,\...
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The map $C^\infty(M,U(A_F))\to C^\infty(M,U(A_F)/\mathfrak{H}(F))$ is an homomorphism.
I am reading "Noncommutative Geometry and Particle Physics" by van Suijlekom. I have problems to identify one map as homomorphism.
Let $M\times F$ be an almost-commutative manifold. The ...
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Smooth vectors for the torus action on the irrational rotation algebra
There exists a canonical action of the group $S^1\times S^1$ on the irrational rotation algebra $A_\theta$, which is the universal C*algebra generated by two unitaries $u$ and $v$ satisfying $uv=e^{i2\...
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Is the preimage of a Hopf subalgebra a Hopf subalgebra?
The following must be simple, but I have no intuition here, so excuse me.
Let $F$ and $G$ be Hopf algebras over a field $k$ (in the usual sence, i.e. Hopf algebras in the category of vector spaces ...
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Concerning the q-deformations of semisimple Lie groups
Recently, I came across a question on the q-analougs of finite groups of Lie type.
Some people say that there is no q-deformations of finite groups in the category of quantum groups which I am not ...
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Are there observables $X_1,\ldots,X_m$ and a state $\rho$ in a Hilbert space $H$ of dimension $n$. Prove that rank$(\text{tr}\ \rho X_iX_j))\le n^2$
This is from the exercise 5.7 of the book An Introduction to Quantum Stochastic Calculus by K.R. Parthasarathy. By observable we mean Hermitian Operators on a Hilbert space $H$ and by state we mean ...
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Non commutative Atiyah-Singer theorem
It is well known that for any unital C*-algebra $A$, $K_0(A)\cong \pi_0(\text{Fred}(\mathcal{H}_A))$ (the non-commutative Atiyah-Jänich theorem). In the monograph "Lectures on Operator Theory&...
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