Non-Commutative Hilbert Nullstellensatz?

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In summary, the Nullstellensatz is a correspondence between commutative rings and algebraic varieties, and a non-commutative space can be defined by starting with a non-commutative algebra.
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"pi"mp
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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!
 
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(nonsense removed)
 
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  • #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?
 
  • #4
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.
 
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  • #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!
 

What is the Non-Commutative Hilbert Nullstellensatz theorem?

The Non-Commutative Hilbert Nullstellensatz theorem is a fundamental result in algebraic geometry that states the connection between ideals in a non-commutative ring and its associated algebraic variety. It is a generalization of the classical Hilbert Nullstellensatz theorem which applies to commutative rings.

What does the Non-Commutative Hilbert Nullstellensatz theorem tell us?

The theorem tells us that given a non-commutative ring R and an ideal I in R, the set of common zeroes of the elements of I is equal to the radical of I. This means that if an element is in the radical of I, then it will be a zero of all the elements in I. Conversely, if an element is a common zero of all the elements in I, then it will be in the radical of I.

How is the Non-Commutative Hilbert Nullstellensatz theorem used in mathematics?

The theorem has many important applications in mathematics, particularly in algebraic geometry and commutative algebra. It is used to study the geometry of algebraic varieties and to prove results about their structure and properties. It also has connections to other areas of mathematics, such as representation theory and non-commutative algebra.

What are some key differences between the Non-Commutative Hilbert Nullstellensatz theorem and the classical Hilbert Nullstellensatz theorem?

The main difference between the two theorems is that the classical Hilbert Nullstellensatz applies to commutative rings, while the Non-Commutative Hilbert Nullstellensatz applies to non-commutative rings. Additionally, the Non-Commutative Hilbert Nullstellensatz involves the concept of radicals, which is not present in the classical version.

What are some other important theorems related to the Non-Commutative Hilbert Nullstellensatz?

Some other important theorems related to the Non-Commutative Hilbert Nullstellensatz include the Artin-Wedderburn theorem, which characterizes semisimple rings, and the Jacobson density theorem, which states that every non-commutative ring contains a maximal commutative subring. Theorems related to the classical Hilbert Nullstellensatz, such as the Nullstellensatz for polynomial rings and the Nullstellensatz for finitely generated algebras, also have connections to the Non-Commutative Hilbert Nullstellensatz.

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