Proving Uniqueness of CCR Algebras

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I am currently trying to learn about CCR algebras (canonical commutation relations) and I am experiencing some confusion with the proof that the CCR algebra of a (non-degenerate, real) symplectic vector space is unique.



I have been reading from chapter 6 of the following, in which the relevant result is Theorem 3:



https://www.math.uni-potsdam.de/fileadmin/user_upload/Prof-Geometrie/Dokumente/Publikationen/qft-alg.pdf



I shall summarise the proof, and insert numbers where I have questions. Any terminology should be defined in the document linked above.



Proof:
Given two CCR-representations ($A_1$, $W_1$) and ($A_2$, $W_2$) of the symplectic vector space ($V$, $omega$), we must show that the $^*$-isomorphism $pi:langle W_1(V)ranglerightarrowlangle W_2(V)rangle$ between the $^*$-algebras $langle W_1(V)rangle$ and $langle W_2(V)rangle$ extends to an isometry between $A_1$ and $A_2$$^(1)$.



Then the norm $|x|=|pi(x)|_2$ is introduced on $A_1$, and it noted that $|pi(x)|_2leq |x|_textmax$, where $|cdot|_textmax$ is defined in Lemma 9 by



$$|x|_textmax=suptext is a C^*text norm on langle W_1(V)rangle.$$



Then it is concluded that $varphi$ extends to a $^*$-homomorphism $overlinelangle W_1(V)rangle^textmaxrightarrow A_2$$^(2)$. Lemma 10 is applied to conclude that this extension is injective, and then it follows that the extension is isometric$^(3)$.



This is pretty much where the proof finishes, barring one note about the case when $A_1=A_2$. I'll now list my questions.



(1) Checking this will be sufficient because we could then do the same for the inverse $varphi^-1$ to get the required map between $A_1$ and $A_2$?



(2) The bound $|pi(x)|_2leq |x|_textmax$ shows that $pi$ is continuous, so $pi$ extends to $overlinelangle W_1(V)rangle^textmax$ by continuity and density of $langle W_1(V)rangle$?



(3) I really don't understand why this is sufficient. To me it seems that finishing at this point requires the completion $overlinelangle W_1(V)rangle^textmax$ to coincide with $A_1$, but I haven't really gotten anywhere with showing this on my own.



Any help/suggestions/slaps around the face because it's obvious will be greatly appreciated.




functional-analysis mathematical-physics operator-algebras




share|cite|improve this question

























    up vote
    5
    down vote

    favorite
    2












    I am currently trying to learn about CCR algebras (canonical commutation relations) and I am experiencing some confusion with the proof that the CCR algebra of a (non-degenerate, real) symplectic vector space is unique.



    I have been reading from chapter 6 of the following, in which the relevant result is Theorem 3:



    https://www.math.uni-potsdam.de/fileadmin/user_upload/Prof-Geometrie/Dokumente/Publikationen/qft-alg.pdf



    I shall summarise the proof, and insert numbers where I have questions. Any terminology should be defined in the document linked above.



    Proof:
    Given two CCR-representations ($A_1$, $W_1$) and ($A_2$, $W_2$) of the symplectic vector space ($V$, $omega$), we must show that the $^*$-isomorphism $pi:langle W_1(V)ranglerightarrowlangle W_2(V)rangle$ between the $^*$-algebras $langle W_1(V)rangle$ and $langle W_2(V)rangle$ extends to an isometry between $A_1$ and $A_2$$^(1)$.



    Then the norm $|x|=|pi(x)|_2$ is introduced on $A_1$, and it noted that $|pi(x)|_2leq |x|_textmax$, where $|cdot|_textmax$ is defined in Lemma 9 by



    $$|x|_textmax=suptext is a C^*text norm on langle W_1(V)rangle.$$



    Then it is concluded that $varphi$ extends to a $^*$-homomorphism $overlinelangle W_1(V)rangle^textmaxrightarrow A_2$$^(2)$. Lemma 10 is applied to conclude that this extension is injective, and then it follows that the extension is isometric$^(3)$.



    This is pretty much where the proof finishes, barring one note about the case when $A_1=A_2$. I'll now list my questions.



    (1) Checking this will be sufficient because we could then do the same for the inverse $varphi^-1$ to get the required map between $A_1$ and $A_2$?



    (2) The bound $|pi(x)|_2leq |x|_textmax$ shows that $pi$ is continuous, so $pi$ extends to $overlinelangle W_1(V)rangle^textmax$ by continuity and density of $langle W_1(V)rangle$?



    (3) I really don't understand why this is sufficient. To me it seems that finishing at this point requires the completion $overlinelangle W_1(V)rangle^textmax$ to coincide with $A_1$, but I haven't really gotten anywhere with showing this on my own.



    Any help/suggestions/slaps around the face because it's obvious will be greatly appreciated.




    functional-analysis mathematical-physics operator-algebras




    share|cite|improve this question























      up vote
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      I am currently trying to learn about CCR algebras (canonical commutation relations) and I am experiencing some confusion with the proof that the CCR algebra of a (non-degenerate, real) symplectic vector space is unique.



      I have been reading from chapter 6 of the following, in which the relevant result is Theorem 3:



      https://www.math.uni-potsdam.de/fileadmin/user_upload/Prof-Geometrie/Dokumente/Publikationen/qft-alg.pdf



      I shall summarise the proof, and insert numbers where I have questions. Any terminology should be defined in the document linked above.



      Proof:
      Given two CCR-representations ($A_1$, $W_1$) and ($A_2$, $W_2$) of the symplectic vector space ($V$, $omega$), we must show that the $^*$-isomorphism $pi:langle W_1(V)ranglerightarrowlangle W_2(V)rangle$ between the $^*$-algebras $langle W_1(V)rangle$ and $langle W_2(V)rangle$ extends to an isometry between $A_1$ and $A_2$$^(1)$.



      Then the norm $|x|=|pi(x)|_2$ is introduced on $A_1$, and it noted that $|pi(x)|_2leq |x|_textmax$, where $|cdot|_textmax$ is defined in Lemma 9 by



      $$|x|_textmax=suptext is a C^*text norm on langle W_1(V)rangle.$$



      Then it is concluded that $varphi$ extends to a $^*$-homomorphism $overlinelangle W_1(V)rangle^textmaxrightarrow A_2$$^(2)$. Lemma 10 is applied to conclude that this extension is injective, and then it follows that the extension is isometric$^(3)$.



      This is pretty much where the proof finishes, barring one note about the case when $A_1=A_2$. I'll now list my questions.



      (1) Checking this will be sufficient because we could then do the same for the inverse $varphi^-1$ to get the required map between $A_1$ and $A_2$?



      (2) The bound $|pi(x)|_2leq |x|_textmax$ shows that $pi$ is continuous, so $pi$ extends to $overlinelangle W_1(V)rangle^textmax$ by continuity and density of $langle W_1(V)rangle$?



      (3) I really don't understand why this is sufficient. To me it seems that finishing at this point requires the completion $overlinelangle W_1(V)rangle^textmax$ to coincide with $A_1$, but I haven't really gotten anywhere with showing this on my own.



      Any help/suggestions/slaps around the face because it's obvious will be greatly appreciated.




      functional-analysis mathematical-physics operator-algebras




      share|cite|improve this question













      I am currently trying to learn about CCR algebras (canonical commutation relations) and I am experiencing some confusion with the proof that the CCR algebra of a (non-degenerate, real) symplectic vector space is unique.



      I have been reading from chapter 6 of the following, in which the relevant result is Theorem 3:



      https://www.math.uni-potsdam.de/fileadmin/user_upload/Prof-Geometrie/Dokumente/Publikationen/qft-alg.pdf



      I shall summarise the proof, and insert numbers where I have questions. Any terminology should be defined in the document linked above.



      Proof:
      Given two CCR-representations ($A_1$, $W_1$) and ($A_2$, $W_2$) of the symplectic vector space ($V$, $omega$), we must show that the $^*$-isomorphism $pi:langle W_1(V)ranglerightarrowlangle W_2(V)rangle$ between the $^*$-algebras $langle W_1(V)rangle$ and $langle W_2(V)rangle$ extends to an isometry between $A_1$ and $A_2$$^(1)$.



      Then the norm $|x|=|pi(x)|_2$ is introduced on $A_1$, and it noted that $|pi(x)|_2leq |x|_textmax$, where $|cdot|_textmax$ is defined in Lemma 9 by



      $$|x|_textmax=suptext is a C^*text norm on langle W_1(V)rangle.$$



      Then it is concluded that $varphi$ extends to a $^*$-homomorphism $overlinelangle W_1(V)rangle^textmaxrightarrow A_2$$^(2)$. Lemma 10 is applied to conclude that this extension is injective, and then it follows that the extension is isometric$^(3)$.



      This is pretty much where the proof finishes, barring one note about the case when $A_1=A_2$. I'll now list my questions.



      (1) Checking this will be sufficient because we could then do the same for the inverse $varphi^-1$ to get the required map between $A_1$ and $A_2$?



      (2) The bound $|pi(x)|_2leq |x|_textmax$ shows that $pi$ is continuous, so $pi$ extends to $overlinelangle W_1(V)rangle^textmax$ by continuity and density of $langle W_1(V)rangle$?



      (3) I really don't understand why this is sufficient. To me it seems that finishing at this point requires the completion $overlinelangle W_1(V)rangle^textmax$ to coincide with $A_1$, but I haven't really gotten anywhere with showing this on my own.



      Any help/suggestions/slaps around the face because it's obvious will be greatly appreciated.





      functional-analysis mathematical-physics operator-algebras





      share|cite|improve this question












      share|cite|improve this question




      share|cite|improve this question








      edited Aug 9 at 14:30
























      asked Jul 18 at 14:36









      user505379

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          I've had a good old think about this, and may have filled in some of the blanks.



          For starters I have some additional confusion in the author's definition of CCR-algebras, they say that if $(A,W)$ is a Weyl system of $(V,omega)$ then $A$ is a CCR-algebra if $A$ is generated as a C$^*$-algebra by the set $W(u)text; uin V$. All texts I have encountered only define C$^*$-subalgebras generated by subsets so I assume that what is meant in this case is that $A$ coincides with the C$^*$-subalgebra of itself generated by $W(u)text; uin V$. If this is what is meant, then it is possible to show using the properties of a Weyl system that $A$ is the norm-closure of the linear span of the set $W(u)text; uin V$.



          Consider the $^*$-isomorphism $pi:langle W_1(V) ranglerightarrowlangle W_2(V) rangle$. The set $langle W_2(V) rangle$ is dense in $A_2$, and the image of the composition



          $$langle W_1(V) rangle oversetpirightarrowlangle W_2(V) rangle oversetihookrightarrow A_2,$$



          (where $i$ denotes inclusion), is $langle W_2(V) rangle$. (Perhaps it is worth noting that this composition is a $*$-homomorphism).



          Defining $|x|:=|pi(x)|_2$ gives a C$^*$-norm on $langle W_1(V) rangle$ so there is the bound $|pi(x)|_2=|x|leq |x|_textmax$ which shows that the composition above is continuous and we may extend it to a map $varphi:overlinelangle W_1(V) rangle^textmaxrightarrow A_2$. Now we invoke Lemma 10 to deduce that $varphi$ is injective and hence isometric.



          Since $varphi$ extends $icirc pi$, the image of $varphi$ must contain $langle W_2(V) rangle$, but since $varphi$ is an isometry into a complete metric space, its image must be closed, therefore $varphi$ is surjective, so is an isomorphism. (Presumably the extension $varphi$ is also a $^*$-homomorphism too).



          I am still a bit stuck on the last part.



          I think that I'm going to take a break from thinking about this, and return in a day or two.






          share|cite|improve this answer























          • Hmm, for the last part I think that it may have been obvious the whole time, but I was just failing to spot it. I think that we just observe that $(overlinelangle W_1(V) rangle^textmax,W)$ satisfies the definition of a CCR-representation, so that $overlinelangle W_1(V) rangle^textmax$ is a CCR algebra. Then replacing $A_2$ by this in the above would give imply the full result.
            – user505379
            Jul 24 at 14:36











          • Well I finished working through this proof, then looked back at their Lemma 10 and realised that I don't understand their argument there either... Oh well.
            – user505379
            Aug 9 at 15:39










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          up vote
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          I've had a good old think about this, and may have filled in some of the blanks.



          For starters I have some additional confusion in the author's definition of CCR-algebras, they say that if $(A,W)$ is a Weyl system of $(V,omega)$ then $A$ is a CCR-algebra if $A$ is generated as a C$^*$-algebra by the set $W(u)text; uin V$. All texts I have encountered only define C$^*$-subalgebras generated by subsets so I assume that what is meant in this case is that $A$ coincides with the C$^*$-subalgebra of itself generated by $W(u)text; uin V$. If this is what is meant, then it is possible to show using the properties of a Weyl system that $A$ is the norm-closure of the linear span of the set $W(u)text; uin V$.



          Consider the $^*$-isomorphism $pi:langle W_1(V) ranglerightarrowlangle W_2(V) rangle$. The set $langle W_2(V) rangle$ is dense in $A_2$, and the image of the composition



          $$langle W_1(V) rangle oversetpirightarrowlangle W_2(V) rangle oversetihookrightarrow A_2,$$



          (where $i$ denotes inclusion), is $langle W_2(V) rangle$. (Perhaps it is worth noting that this composition is a $*$-homomorphism).



          Defining $|x|:=|pi(x)|_2$ gives a C$^*$-norm on $langle W_1(V) rangle$ so there is the bound $|pi(x)|_2=|x|leq |x|_textmax$ which shows that the composition above is continuous and we may extend it to a map $varphi:overlinelangle W_1(V) rangle^textmaxrightarrow A_2$. Now we invoke Lemma 10 to deduce that $varphi$ is injective and hence isometric.



          Since $varphi$ extends $icirc pi$, the image of $varphi$ must contain $langle W_2(V) rangle$, but since $varphi$ is an isometry into a complete metric space, its image must be closed, therefore $varphi$ is surjective, so is an isomorphism. (Presumably the extension $varphi$ is also a $^*$-homomorphism too).



          I am still a bit stuck on the last part.



          I think that I'm going to take a break from thinking about this, and return in a day or two.






          share|cite|improve this answer























          • Hmm, for the last part I think that it may have been obvious the whole time, but I was just failing to spot it. I think that we just observe that $(overlinelangle W_1(V) rangle^textmax,W)$ satisfies the definition of a CCR-representation, so that $overlinelangle W_1(V) rangle^textmax$ is a CCR algebra. Then replacing $A_2$ by this in the above would give imply the full result.
            – user505379
            Jul 24 at 14:36











          • Well I finished working through this proof, then looked back at their Lemma 10 and realised that I don't understand their argument there either... Oh well.
            – user505379
            Aug 9 at 15:39














          up vote
          0
          down vote













          I've had a good old think about this, and may have filled in some of the blanks.



          For starters I have some additional confusion in the author's definition of CCR-algebras, they say that if $(A,W)$ is a Weyl system of $(V,omega)$ then $A$ is a CCR-algebra if $A$ is generated as a C$^*$-algebra by the set $W(u)text; uin V$. All texts I have encountered only define C$^*$-subalgebras generated by subsets so I assume that what is meant in this case is that $A$ coincides with the C$^*$-subalgebra of itself generated by $W(u)text; uin V$. If this is what is meant, then it is possible to show using the properties of a Weyl system that $A$ is the norm-closure of the linear span of the set $W(u)text; uin V$.



          Consider the $^*$-isomorphism $pi:langle W_1(V) ranglerightarrowlangle W_2(V) rangle$. The set $langle W_2(V) rangle$ is dense in $A_2$, and the image of the composition



          $$langle W_1(V) rangle oversetpirightarrowlangle W_2(V) rangle oversetihookrightarrow A_2,$$



          (where $i$ denotes inclusion), is $langle W_2(V) rangle$. (Perhaps it is worth noting that this composition is a $*$-homomorphism).



          Defining $|x|:=|pi(x)|_2$ gives a C$^*$-norm on $langle W_1(V) rangle$ so there is the bound $|pi(x)|_2=|x|leq |x|_textmax$ which shows that the composition above is continuous and we may extend it to a map $varphi:overlinelangle W_1(V) rangle^textmaxrightarrow A_2$. Now we invoke Lemma 10 to deduce that $varphi$ is injective and hence isometric.



          Since $varphi$ extends $icirc pi$, the image of $varphi$ must contain $langle W_2(V) rangle$, but since $varphi$ is an isometry into a complete metric space, its image must be closed, therefore $varphi$ is surjective, so is an isomorphism. (Presumably the extension $varphi$ is also a $^*$-homomorphism too).



          I am still a bit stuck on the last part.



          I think that I'm going to take a break from thinking about this, and return in a day or two.






          share|cite|improve this answer























          • Hmm, for the last part I think that it may have been obvious the whole time, but I was just failing to spot it. I think that we just observe that $(overlinelangle W_1(V) rangle^textmax,W)$ satisfies the definition of a CCR-representation, so that $overlinelangle W_1(V) rangle^textmax$ is a CCR algebra. Then replacing $A_2$ by this in the above would give imply the full result.
            – user505379
            Jul 24 at 14:36











          • Well I finished working through this proof, then looked back at their Lemma 10 and realised that I don't understand their argument there either... Oh well.
            – user505379
            Aug 9 at 15:39












          up vote
          0
          down vote










          up vote
          0
          down vote









          I've had a good old think about this, and may have filled in some of the blanks.



          For starters I have some additional confusion in the author's definition of CCR-algebras, they say that if $(A,W)$ is a Weyl system of $(V,omega)$ then $A$ is a CCR-algebra if $A$ is generated as a C$^*$-algebra by the set $W(u)text; uin V$. All texts I have encountered only define C$^*$-subalgebras generated by subsets so I assume that what is meant in this case is that $A$ coincides with the C$^*$-subalgebra of itself generated by $W(u)text; uin V$. If this is what is meant, then it is possible to show using the properties of a Weyl system that $A$ is the norm-closure of the linear span of the set $W(u)text; uin V$.



          Consider the $^*$-isomorphism $pi:langle W_1(V) ranglerightarrowlangle W_2(V) rangle$. The set $langle W_2(V) rangle$ is dense in $A_2$, and the image of the composition



          $$langle W_1(V) rangle oversetpirightarrowlangle W_2(V) rangle oversetihookrightarrow A_2,$$



          (where $i$ denotes inclusion), is $langle W_2(V) rangle$. (Perhaps it is worth noting that this composition is a $*$-homomorphism).



          Defining $|x|:=|pi(x)|_2$ gives a C$^*$-norm on $langle W_1(V) rangle$ so there is the bound $|pi(x)|_2=|x|leq |x|_textmax$ which shows that the composition above is continuous and we may extend it to a map $varphi:overlinelangle W_1(V) rangle^textmaxrightarrow A_2$. Now we invoke Lemma 10 to deduce that $varphi$ is injective and hence isometric.



          Since $varphi$ extends $icirc pi$, the image of $varphi$ must contain $langle W_2(V) rangle$, but since $varphi$ is an isometry into a complete metric space, its image must be closed, therefore $varphi$ is surjective, so is an isomorphism. (Presumably the extension $varphi$ is also a $^*$-homomorphism too).



          I am still a bit stuck on the last part.



          I think that I'm going to take a break from thinking about this, and return in a day or two.






          share|cite|improve this answer















          I've had a good old think about this, and may have filled in some of the blanks.



          For starters I have some additional confusion in the author's definition of CCR-algebras, they say that if $(A,W)$ is a Weyl system of $(V,omega)$ then $A$ is a CCR-algebra if $A$ is generated as a C$^*$-algebra by the set $W(u)text; uin V$. All texts I have encountered only define C$^*$-subalgebras generated by subsets so I assume that what is meant in this case is that $A$ coincides with the C$^*$-subalgebra of itself generated by $W(u)text; uin V$. If this is what is meant, then it is possible to show using the properties of a Weyl system that $A$ is the norm-closure of the linear span of the set $W(u)text; uin V$.



          Consider the $^*$-isomorphism $pi:langle W_1(V) ranglerightarrowlangle W_2(V) rangle$. The set $langle W_2(V) rangle$ is dense in $A_2$, and the image of the composition



          $$langle W_1(V) rangle oversetpirightarrowlangle W_2(V) rangle oversetihookrightarrow A_2,$$



          (where $i$ denotes inclusion), is $langle W_2(V) rangle$. (Perhaps it is worth noting that this composition is a $*$-homomorphism).



          Defining $|x|:=|pi(x)|_2$ gives a C$^*$-norm on $langle W_1(V) rangle$ so there is the bound $|pi(x)|_2=|x|leq |x|_textmax$ which shows that the composition above is continuous and we may extend it to a map $varphi:overlinelangle W_1(V) rangle^textmaxrightarrow A_2$. Now we invoke Lemma 10 to deduce that $varphi$ is injective and hence isometric.



          Since $varphi$ extends $icirc pi$, the image of $varphi$ must contain $langle W_2(V) rangle$, but since $varphi$ is an isometry into a complete metric space, its image must be closed, therefore $varphi$ is surjective, so is an isomorphism. (Presumably the extension $varphi$ is also a $^*$-homomorphism too).



          I am still a bit stuck on the last part.



          I think that I'm going to take a break from thinking about this, and return in a day or two.







          share|cite|improve this answer















          share|cite|improve this answer



          share|cite|improve this answer








          edited Jul 20 at 16:59


























          answered Jul 19 at 15:35









          user505379

          9019




          9019











          • Hmm, for the last part I think that it may have been obvious the whole time, but I was just failing to spot it. I think that we just observe that $(overlinelangle W_1(V) rangle^textmax,W)$ satisfies the definition of a CCR-representation, so that $overlinelangle W_1(V) rangle^textmax$ is a CCR algebra. Then replacing $A_2$ by this in the above would give imply the full result.
            – user505379
            Jul 24 at 14:36











          • Well I finished working through this proof, then looked back at their Lemma 10 and realised that I don't understand their argument there either... Oh well.
            – user505379
            Aug 9 at 15:39
















          • Hmm, for the last part I think that it may have been obvious the whole time, but I was just failing to spot it. I think that we just observe that $(overlinelangle W_1(V) rangle^textmax,W)$ satisfies the definition of a CCR-representation, so that $overlinelangle W_1(V) rangle^textmax$ is a CCR algebra. Then replacing $A_2$ by this in the above would give imply the full result.
            – user505379
            Jul 24 at 14:36











          • Well I finished working through this proof, then looked back at their Lemma 10 and realised that I don't understand their argument there either... Oh well.
            – user505379
            Aug 9 at 15:39















          Hmm, for the last part I think that it may have been obvious the whole time, but I was just failing to spot it. I think that we just observe that $(overlinelangle W_1(V) rangle^textmax,W)$ satisfies the definition of a CCR-representation, so that $overlinelangle W_1(V) rangle^textmax$ is a CCR algebra. Then replacing $A_2$ by this in the above would give imply the full result.
          – user505379
          Jul 24 at 14:36





          Hmm, for the last part I think that it may have been obvious the whole time, but I was just failing to spot it. I think that we just observe that $(overlinelangle W_1(V) rangle^textmax,W)$ satisfies the definition of a CCR-representation, so that $overlinelangle W_1(V) rangle^textmax$ is a CCR algebra. Then replacing $A_2$ by this in the above would give imply the full result.
          – user505379
          Jul 24 at 14:36













          Well I finished working through this proof, then looked back at their Lemma 10 and realised that I don't understand their argument there either... Oh well.
          – user505379
          Aug 9 at 15:39




          Well I finished working through this proof, then looked back at their Lemma 10 and realised that I don't understand their argument there either... Oh well.
          – user505379
          Aug 9 at 15:39












           

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          What is the equation of a 3D cone with generalised tilt?

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          Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite What is the equation of a 3D cone with generalized tilt? I've noticed that in most equations given to represent a cone, there is no parameter which defines the tilt of the cone in 3D space and that most of them have their point at the origin $(0,0,0)$ - I was wondering if anyone could give me a more generalized cone equation for a cone in any position in 3D space 3d share | cite | improve this question edited Jul 25 '16 at 0:05 ervx 9,425 3 13 36 asked Jul 24 '16 at 15:38 Charlene 11 2 2 Welcome to Math.SE! Could you please explain a few things to help people give an answer useful to you: 1. What do you mean by "generalized tilt"? 2. Are you asking about a right circular cone? Does the angle at the vertex matter? 3. What form of answer (implicit, parametric...?) are you looking for? 4. If this is homework, what tools do you have available, an...
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          Relationship between determinant of matrix and determinant of adjoint?

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          Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite We are studying adjoints in class, and I was curious if there is a relationship between the determinant of matrix A, and the determinant of the adjoint of matrix A? I assume there would be a relationship because finding the adjoint requires creating a cofactor matrix and then transposing it. linear-algebra determinant adjoint-operators share | cite | improve this question asked Nov 17 '17 at 17:32 CluelessCoder 32 5 For a square matrix $A$ of order $n$, we have $A(operatornameadj A)=(operatornameadj A)A=|A|I_n$ where $I_n$ is the identity matrix of order $n$. This should tell you about the determinant of the adjoint in terms of that of $A$ (use multiplicative property and other elementary properties of determinants). – Prasun Biswas Nov 17 '17 at 17:35 1 @CluelessCoder: see here: math.stackexchange.com/questions/516127/…...
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          Color the edges and diagonals of a regular polygon

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          Clash Royale CLAN TAG #URR8PPP up vote 5 down vote favorite 1 Here is the problem: For what $n$ is it possible to color the edges and diagonals of an $n$-side regular polygon with $dfracbinomn23$ colors, such that you use every color exactly three times and for every color the three segments (edges or diagonals) with that color form a triangle? Trivially $nequiv 0 text or 1 pmod3$. I can also prove that if the statement is true for $k$ than it is true for $3k$ as well. How to finish? Please help! Thanks combinatorial-geometry share | cite | improve this question edited Aug 6 at 21:57 alcana 118 4 asked Aug 6 at 21:15 Leo Gardner 356 11 Even assuming "polyhpn" in the title is a typo for polygon , it is unclear what you mean by "the edges and sides". Aren't the side of a regular polygon the same as the edges? A regular $n$-sided polygon has $n$ edges. – hardmath Aug 6 at 21:20 3 $n$ ne...
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