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Jun
8
comment 8 periodicity: Clifford clock- Bott periodicity - KO-dimension in noncommutative geometries
For spin manifolds, the canonical Dirac operator $D\!\!\!\!\!{/}$ is allways elliptic. $D\!\!\!\!\!{/}$, whose square has leading term the inverse metric, would lose its ellipticity if considered over a pseudo-Riemannian manifold, for it would vanish outside the zero section, precisely on "light-like vectors". By the way, in the noncommutative world, I can no longer see analytic facts so apart from algebraic and geometric facts - in a sense, analysis, geometry and many things are encoded only in algebra.
Jun
8
comment 8 periodicity: Clifford clock- Bott periodicity - KO-dimension in noncommutative geometries
I agree, the KO-dim is defined enterely in algebraic terms; I never said the contrary. What I said is that, nevertheless, if you understand KO-dim of spectral triples as being related to the signature of the "underlying Riemannian manifold" -as you kindly pointed out- you run into trouble. To see that,there are ordinary Riemannian spin manifolds having various KO-dimensions.
Jun
8
comment 8 periodicity: Clifford clock- Bott periodicity - KO-dimension in noncommutative geometries
Thank you Paul. I see what you say. However, I do not agree in the last sentence of your first comment. There is no noncommutative analogue of pseudo-Riemannian manifolds. Basically if you let signatures other than $(+,...+)$ you lose the ellipticity of the Dirac operator.
Jun
8
answered Riemann tensor on a sphere
Jun
7
revised 8 periodicity: Clifford clock- Bott periodicity - KO-dimension in noncommutative geometries
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Jun
7
revised 8 periodicity: Clifford clock- Bott periodicity - KO-dimension in noncommutative geometries
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Jun
7
revised 8 periodicity: Clifford clock- Bott periodicity - KO-dimension in noncommutative geometries
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Jun
7
comment 8 periodicity: Clifford clock- Bott periodicity - KO-dimension in noncommutative geometries
I thought it was a very soft-question for people familiar with K-theory, but what you say sounds very reasonable. I'll reedit the question.
Jun
7
revised 8 periodicity: Clifford clock- Bott periodicity - KO-dimension in noncommutative geometries
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Jun
7
comment 8 periodicity: Clifford clock- Bott periodicity - KO-dimension in noncommutative geometries
Thank you for the reference. I restated the question. Now, Connes doesn't quite explain why the dimension he introduces for spectral triples has a K-theoretical meaning. It is not obvious for non-K-theorists, at least.
Jun
7
awarded  Editor
Jun
7
revised 8 periodicity: Clifford clock- Bott periodicity - KO-dimension in noncommutative geometries
edited body
Jun
7
asked 8 periodicity: Clifford clock- Bott periodicity - KO-dimension in noncommutative geometries
Jun
6
accepted #(cardinality of irreps Lie algebras) > #(irreps of ssociative algebras). Proof?
Jun
6
comment #(cardinality of irreps Lie algebras) > #(irreps of ssociative algebras). Proof?
Since $A\cong\oplus_{i=0}^{N}M_{n_i}(\mathbb{C})$ for any fininte dimensional $*$-algebra $A$, and any matrix algebra direct summand has a unique irrep, to wit $\mathbf{n}=\mathbb{C}^n$ acted on by the left, I get an "obvious" representation for $A$ in the sum $V:=\oplus_i^N \mathbb{n}_{i}$. It seems to me pretty canonical-but, as I told you, I am not sure if that is the term I should use. The point is that I cannot do the same for semisimple Lie algebras, for their classification is quite more complicated, can I?Sorry to ask again, but could you shed some light on the bijection you mentioned?
Jun
5
comment #(cardinality of irreps Lie algebras) > #(irreps of ssociative algebras). Proof?
by canonical I meant, that if I have a $*$-algebra, I can get an canonical representation in the sum of all its irreps. I'm not sure of having used the right word though. Can you shed some light on the bijection you mentioned?
Jun
5
answered Notation for antiderivative
Jun
5
comment #(cardinality of irreps Lie algebras) > #(irreps of ssociative algebras). Proof?
So, given a Lie algebra, you cannot form a canonical representation, can you? (unlike the $*$-algebra case).
Jun
5
awarded  Scholar
Jun
5
awarded  Student