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Non-Commutative Hilbert Nullstellensatz?

  1. May 31, 2015 #1
    Hi,
    In my very naive understanding of algebraic geometry, I get the impression that it's written in language of commutative algebra and the main theorem (at least at the basic level) is Hilbert's Nullstellensatz. I'm curious if there's an analog of the Nullstellensatz for non-commutative algebra/geometry?

    What I'm envisioning would be a correspondence between some operator/matrix algebra and a "fuzzy" geometry.
    Thanks!
     
  2. jcsd
  3. May 31, 2015 #2

    wabbit

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    (nonsense removed)
     
    Last edited: May 31, 2015
  4. May 31, 2015 #3
    Thanks! But I still have some confusions:

    --I had never thought of the Nullstellensatz as having to do with functions defined on a variety. I had thought it was just a correspondence between commutative rings and algebraic varieties. Is there a direct analog of the vanishing set of polynomials in the non-commutative case?

    --Also, you talk about defining a non-commutative space by starting with a non-commutative algebra as the functions on the space. But I vaguely remember reading somewhere that the problem with this is that not all non-commutative geometries will have functions defined on them that form a non-commutative algebra. Is this wrong?
     
  5. May 31, 2015 #4

    wabbit

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    I think.you are right, I took too much of a shortcut here. Interesting question, need to think about this.

    Edit : actually, what I said is just wrong, sorry about that.
     
    Last edited: May 31, 2015
  6. May 31, 2015 #5
    No worries, thank you. It seems like the standard procedure is to start with some non-commutative algebra, like the Heisenberg algebra or the angular momentum algebra, and define a "symbol map" that associates an element in this algebra to a function on the ordinary commutative manifold. Now, from what I read this certainly isn't a unique map, but I don't even understand how it can be well defined! It seems like you need to know before hand what the fuzzy geometry is going to be in order to use this. i.e. you need to know that the Heisenberg algebra will yield the non-commutative plane and that the angular momentum SU(2) algebra will yield the non-commutative sphere.

    I'm not very clever with mathematical details, but I can usually appreciate the general program of some field. This I'm completely lost on!
     
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