Skip to main content
MathOverflow

Return to Question

added 377 characters in body
Source Link
Tim Campion
Tim Campion
  • 64k
  • 13
  • 143
  • 384

The automorphism group $Aut(E_1)$ of the $E_1$ operad is the cyclic group of order 2, $C_2$, and thus $C_2$ acts on any category of algebras (by reversing the multiplication). The seeming coincidence that $Gal(\mathbb C/\mathbb R) = C_2$ as well allows one to define such notions as $C^\ast$ algebras, where one asks for an appropriate interaction between the actions of $Gal(\mathbb C / \mathbb R)$ and $Aut(E_1)$.

It goes without saying that by the law of small numbers, many mathematical objects will have automorphism group $C_2$, and in general one shouldn't expect to read too much into such coincidences. But the simple fact that $C^\ast$-algebras and other sorts of $\mathbb C$-algebras with involution turn out to be so incredibly useful in mathematics (and physics, for that matter) suggests to me that maybe in this case, it's not a coincidence that $Gal(\mathbb C/\mathbb R) = Aut(E_1)$. However, I can't really think of a "good reason" that these two groups are isomorphic. So I'll ask

Questions:

  1. Is there some "good reason" why $Gal(\mathbb C / \mathbb R)$ acts on the category of $\mathbb C$-algebras in two different ways -- both canonically and by reversing multiplication?

  2. Even better, is there a "good reason" why $Gal(\mathbb C / \mathbb R)$ acts on the $E_1$ operad?

  3. Is there some generalization of the notion of algebra with involution that makes sense for other Galois extensions? Or is there another direction of generalization which sheds light on the usefulness of $\mathbb C$-algebras with involution?

  4. If there's not really a "good reason" for all of this, then is there some other way to explain the usefulness of the notion of a $C^\ast$ algebra, or a $\mathbb C$-algebra with involution?

In (1) and (2), roughly by a "good reason" for an action I mean: could you construct this action without having done the computation to show that $Gal(\mathbb C / \mathbb R) = C_2$.

If I recall correctly, the fact that $Gal(\mathbb C / \mathbb R) = C_2$ could be chalked up to the fact (I think?) that any finite-index subfield of an algebraically closed field is of index 2. But (perhaps since I'm not familiar with the proof of this fact) I'm not sure how to relate this to the idea that one can reverse the multiplication on an algebra to get another algebra.

I'm tempted to frame this in the more general context of the notion of duality in mathematics, and to ask "Which mathematical dualities are "related", and which ones are "unrelated"?" -- but I think I may need to sharpen the question a bit before asking something like that here on MO.

The automorphism group $Aut(E_1)$ of the $E_1$ operad is the cyclic group of order 2, $C_2$, and thus $C_2$ acts on any category of algebras (by reversing the multiplication). The seeming coincidence that $Gal(\mathbb C/\mathbb R) = C_2$ as well allows one to define such notions as $C^\ast$ algebras, where one asks for an appropriate interaction between the actions of $Gal(\mathbb C / \mathbb R)$ and $Aut(E_1)$.

It goes without saying that by the law of small numbers, many mathematical objects will have automorphism group $C_2$, and in general one shouldn't expect to read too much into such coincidences. But the simple fact that $C^\ast$-algebras and other sorts of $\mathbb C$-algebras with involution turn out to be so incredibly useful in mathematics (and physics, for that matter) suggests to me that maybe in this case, it's not a coincidence that $Gal(\mathbb C/\mathbb R) = Aut(E_1)$. However, I can't really think of a "good reason" that these two groups are isomorphic. So I'll ask

Questions:

  1. Is there some "good reason" why $Gal(\mathbb C / \mathbb R)$ acts on the category of $\mathbb C$-algebras in two different ways -- both canonically and by reversing multiplication?

  2. Even better, is there a "good reason" why $Gal(\mathbb C / \mathbb R)$ acts on the $E_1$ operad?

  3. Is there some generalization of the notion of algebra with involution that makes sense for other Galois extensions? Or is there another direction of generalization which sheds light on the usefulness of $\mathbb C$-algebras with involution?

  4. If there's not really a "good reason" for all of this, then is there some other way to explain the usefulness of the notion of a $C^\ast$ algebra, or a $\mathbb C$-algebra with involution?

In (1) and (2), roughly by a "good reason" for an action I mean: could you construct this action without having done the computation to show that $Gal(\mathbb C / \mathbb R) = C_2$.

I'm tempted to frame this in the more general context of the notion of duality in mathematics, and to ask "Which mathematical dualities are "related", and which ones are "unrelated"?" -- but I think I may need to sharpen the question a bit before asking something like that here on MO.

The automorphism group $Aut(E_1)$ of the $E_1$ operad is the cyclic group of order 2, $C_2$, and thus $C_2$ acts on any category of algebras (by reversing the multiplication). The seeming coincidence that $Gal(\mathbb C/\mathbb R) = C_2$ as well allows one to define such notions as $C^\ast$ algebras, where one asks for an appropriate interaction between the actions of $Gal(\mathbb C / \mathbb R)$ and $Aut(E_1)$.

It goes without saying that by the law of small numbers, many mathematical objects will have automorphism group $C_2$, and in general one shouldn't expect to read too much into such coincidences. But the simple fact that $C^\ast$-algebras and other sorts of $\mathbb C$-algebras with involution turn out to be so incredibly useful in mathematics (and physics, for that matter) suggests to me that maybe in this case, it's not a coincidence that $Gal(\mathbb C/\mathbb R) = Aut(E_1)$. However, I can't really think of a "good reason" that these two groups are isomorphic. So I'll ask

Questions:

  1. Is there some "good reason" why $Gal(\mathbb C / \mathbb R)$ acts on the category of $\mathbb C$-algebras in two different ways -- both canonically and by reversing multiplication?

  2. Even better, is there a "good reason" why $Gal(\mathbb C / \mathbb R)$ acts on the $E_1$ operad?

  3. Is there some generalization of the notion of algebra with involution that makes sense for other Galois extensions? Or is there another direction of generalization which sheds light on the usefulness of $\mathbb C$-algebras with involution?

  4. If there's not really a "good reason" for all of this, then is there some other way to explain the usefulness of the notion of a $C^\ast$ algebra, or a $\mathbb C$-algebra with involution?

In (1) and (2), roughly by a "good reason" for an action I mean: could you construct this action without having done the computation to show that $Gal(\mathbb C / \mathbb R) = C_2$.

If I recall correctly, the fact that $Gal(\mathbb C / \mathbb R) = C_2$ could be chalked up to the fact (I think?) that any finite-index subfield of an algebraically closed field is of index 2. But (perhaps since I'm not familiar with the proof of this fact) I'm not sure how to relate this to the idea that one can reverse the multiplication on an algebra to get another algebra.

I'm tempted to frame this in the more general context of the notion of duality in mathematics, and to ask "Which mathematical dualities are "related", and which ones are "unrelated"?" -- but I think I may need to sharpen the question a bit before asking something like that here on MO.

Source Link
Tim Campion
Tim Campion
  • 64k
  • 13
  • 143
  • 384

Is it a coincidence that $Gal(\mathbb C / \mathbb R) \cong C_2 \cong Aut(E_1)$? (Or: why are $\mathbb C$-algebras with involution so useful?)

The automorphism group $Aut(E_1)$ of the $E_1$ operad is the cyclic group of order 2, $C_2$, and thus $C_2$ acts on any category of algebras (by reversing the multiplication). The seeming coincidence that $Gal(\mathbb C/\mathbb R) = C_2$ as well allows one to define such notions as $C^\ast$ algebras, where one asks for an appropriate interaction between the actions of $Gal(\mathbb C / \mathbb R)$ and $Aut(E_1)$.

It goes without saying that by the law of small numbers, many mathematical objects will have automorphism group $C_2$, and in general one shouldn't expect to read too much into such coincidences. But the simple fact that $C^\ast$-algebras and other sorts of $\mathbb C$-algebras with involution turn out to be so incredibly useful in mathematics (and physics, for that matter) suggests to me that maybe in this case, it's not a coincidence that $Gal(\mathbb C/\mathbb R) = Aut(E_1)$. However, I can't really think of a "good reason" that these two groups are isomorphic. So I'll ask

Questions:

  1. Is there some "good reason" why $Gal(\mathbb C / \mathbb R)$ acts on the category of $\mathbb C$-algebras in two different ways -- both canonically and by reversing multiplication?

  2. Even better, is there a "good reason" why $Gal(\mathbb C / \mathbb R)$ acts on the $E_1$ operad?

  3. Is there some generalization of the notion of algebra with involution that makes sense for other Galois extensions? Or is there another direction of generalization which sheds light on the usefulness of $\mathbb C$-algebras with involution?

  4. If there's not really a "good reason" for all of this, then is there some other way to explain the usefulness of the notion of a $C^\ast$ algebra, or a $\mathbb C$-algebra with involution?

In (1) and (2), roughly by a "good reason" for an action I mean: could you construct this action without having done the computation to show that $Gal(\mathbb C / \mathbb R) = C_2$.

I'm tempted to frame this in the more general context of the notion of duality in mathematics, and to ask "Which mathematical dualities are "related", and which ones are "unrelated"?" -- but I think I may need to sharpen the question a bit before asking something like that here on MO.