Mixing
Representations of the $mathfrak{su}(2)$ Lie algebra in the product space $Votimes Votimes V$
Let $V=mathbb{C}^d$ be a $d$-dimensional vector space over $mathbb{C}$. The problem is to find all possible representations $rho:mathfrak{l}to mathrm{End}(V^3)$ of the Lie algebra $mathfrak{l}=mathfrak{su}(2)$ in the product space $V^3=Votimes Votimes V$ with the following property:
(*)There exists two elements $l_1,l_2in mathfrak{l}$ such that $rho(l_1)=A_{12}+Cequiv Aotimes mathbb{1}+C,rho(l_2)=A_{23}+Cequiv mathbb{1}otimes A +C$, where $Ain mathrm{End}(V^2)$ (i.e. acts in $Votimes V$), $mathbb{1}$ is the identity operator in $mathrm{End}(V)$, and $A_{12},A_{23}$ denotes two different ways of embedding of $A$ in $mathrm{End}(V^3)$. $Cin mathrm{End}(V^3)$ is a suitable "central element" that commutes with $rho(l)$ for $forall lin mathfrak{l}$.
Examples:
(1). For $d$=2, consider the representation $rho(s^x)=sigma^z_1 sigma^x_2/2,rho(s^z)=sigma^z_2 sigma^x_3/2,rho(s^y)=sigma^z_1 sigma^y_2sigma^x_3/2$, where $sigma^x,sigma^y,sigma^z$ are Pauli matrices. This is the 8-dimensional direct sum representation $frac{1}{2}oplusfrac{1}{2}oplusfrac{1}{2}oplusfrac{1}{2}$. We have $l_1=s^x,l_2=s^z,A=sigma^zotimessigma^x$ (or $A_{12}=sigma^z_1sigma^x_2,A_{23}=sigma^z_2sigma^x_3$), and $C=0$. So this representation satisfies the above property(*).
(2) For arbitrary $d$, consider the representation
begin{eqnarray}
rho(s^x)=frac{2P_{12}-P_{23}-P_{13}}{3sqrt{6}}+frac{M}{6i}\
rho(s^y)=frac{2P_{23}-P_{13}-P_{12}}{3sqrt{6}}+frac{M}{6i}\
rho(s^z)=frac{2P_{13}-P_{12}-P_{23}}{3sqrt{6}}+frac{M}{6i},
end{eqnarray}
where $Pinmathrm{End}(V^2)$ is the permutation operator defined by $P(e_1otimes e_2)=e_2otimes e_1$ for $forall e_1,e_2in V$, and $P_{12},P_{23},P_{13}$ are the different embedding of $P$ in $mathrm{End}(V^3)$, e.g. $P_{13}(e_1otimes e_2otimes e_3)=(e_3otimes e_2otimes e_1)$, etc., and $M=[P_{12},P_{23}]=P_{12}P_{23}-P_{23}P_{12}=[P_{23},P_{13}]=[P_{13},P_{12}]$. Put it more formally, $rho$ maps $mathfrak{su}(2)$ to the group algebra $mathbb{C}(S_3)$ of the symmetry group $S_3$. Therefore, let $l_1=frac{2s^x-s^y-s^z}{sqrt{6}},l_2=frac{2s^y-s^z-s^x}{sqrt{6}}$, we have $$rho(l_1)=frac{P_{12}}{2}-frac{c}{6},~~~rho(l_2)=frac{P_{23}}{2}-frac{c}{6},$$ where $c=P_{12}+P_{23}+P_{13}$ is a central element of $mathbb{C}(S_3)$ and therefore commutes with $rho(l)$ for $forall lin mathfrak{su}(2)$. Therefore, this representation satisfies the property(*) with $A=P/2$ and $C=-c/6$. When $d=2$, this representation is the 8-dimensional direct sum representation $0^4oplus frac{1}{2}oplusfrac{1}{2}$.
The above two are all the examples I know up to now. Is there a systematic way of finding all such representations? Interestingly, notice the angle between $l_1$ and $l_2$: in example(1), $angle(l_1,l_2)=pi/2$, in example(2), $angle(l_1,l_2)=2pi/3$. What are all possible allowed angles between $l_1$ and $l_2$ (infinitely many or finitely many)? Or are $pi/2,2pi/3$ the only allowed values of $angle(l_1,l_2)$?
Here is an alternative formulation of the same problem: find all operators $Ain mathrm{End}(V^2)$ such that the set of operators ${A_{12}+C,A_{23}+C}$ in $mathrm{End}(V^3)$ generates the Lie algebra $mathfrak{su}(2)$, for a suitable $Cin mathrm{End}(V^3)$ that commutes with $A_{12},A_{23}$. In example (1) $A=sigma^zotimessigma^x,C=0$ and in example (2) $A=P/2,C=-(P_{12}+P_{23}+P_{13})/6$.
Any hints, references, or further examples will be welcomed. Thanks!
representation-theory lie-algebras
asked Nov 20 '18 at 18:28
Lagrenge
1208
add a comment |
Let $V=mathbb{C}^d$ be a $d$-dimensional vector space over $mathbb{C}$. The problem is to find all possible representations $rho:mathfrak{l}to mathrm{End}(V^3)$ of the Lie algebra $mathfrak{l}=mathfrak{su}(2)$ in the product space $V^3=Votimes Votimes V$ with the following property:
(*)There exists two elements $l_1,l_2in mathfrak{l}$ such that $rho(l_1)=A_{12}+Cequiv Aotimes mathbb{1}+C,rho(l_2)=A_{23}+Cequiv mathbb{1}otimes A +C$, where $Ain mathrm{End}(V^2)$ (i.e. acts in $Votimes V$), $mathbb{1}$ is the identity operator in $mathrm{End}(V)$, and $A_{12},A_{23}$ denotes two different ways of embedding of $A$ in $mathrm{End}(V^3)$. $Cin mathrm{End}(V^3)$ is a suitable "central element" that commutes with $rho(l)$ for $forall lin mathfrak{l}$.
Examples:
(1). For $d$=2, consider the representation $rho(s^x)=sigma^z_1 sigma^x_2/2,rho(s^z)=sigma^z_2 sigma^x_3/2,rho(s^y)=sigma^z_1 sigma^y_2sigma^x_3/2$, where $sigma^x,sigma^y,sigma^z$ are Pauli matrices. This is the 8-dimensional direct sum representation $frac{1}{2}oplusfrac{1}{2}oplusfrac{1}{2}oplusfrac{1}{2}$. We have $l_1=s^x,l_2=s^z,A=sigma^zotimessigma^x$ (or $A_{12}=sigma^z_1sigma^x_2,A_{23}=sigma^z_2sigma^x_3$), and $C=0$. So this representation satisfies the above property(*).
(2) For arbitrary $d$, consider the representation
begin{eqnarray}
rho(s^x)=frac{2P_{12}-P_{23}-P_{13}}{3sqrt{6}}+frac{M}{6i}\
rho(s^y)=frac{2P_{23}-P_{13}-P_{12}}{3sqrt{6}}+frac{M}{6i}\
rho(s^z)=frac{2P_{13}-P_{12}-P_{23}}{3sqrt{6}}+frac{M}{6i},
end{eqnarray}
where $Pinmathrm{End}(V^2)$ is the permutation operator defined by $P(e_1otimes e_2)=e_2otimes e_1$ for $forall e_1,e_2in V$, and $P_{12},P_{23},P_{13}$ are the different embedding of $P$ in $mathrm{End}(V^3)$, e.g. $P_{13}(e_1otimes e_2otimes e_3)=(e_3otimes e_2otimes e_1)$, etc., and $M=[P_{12},P_{23}]=P_{12}P_{23}-P_{23}P_{12}=[P_{23},P_{13}]=[P_{13},P_{12}]$. Put it more formally, $rho$ maps $mathfrak{su}(2)$ to the group algebra $mathbb{C}(S_3)$ of the symmetry group $S_3$. Therefore, let $l_1=frac{2s^x-s^y-s^z}{sqrt{6}},l_2=frac{2s^y-s^z-s^x}{sqrt{6}}$, we have $$rho(l_1)=frac{P_{12}}{2}-frac{c}{6},~~~rho(l_2)=frac{P_{23}}{2}-frac{c}{6},$$ where $c=P_{12}+P_{23}+P_{13}$ is a central element of $mathbb{C}(S_3)$ and therefore commutes with $rho(l)$ for $forall lin mathfrak{su}(2)$. Therefore, this representation satisfies the property(*) with $A=P/2$ and $C=-c/6$. When $d=2$, this representation is the 8-dimensional direct sum representation $0^4oplus frac{1}{2}oplusfrac{1}{2}$.
The above two are all the examples I know up to now. Is there a systematic way of finding all such representations? Interestingly, notice the angle between $l_1$ and $l_2$: in example(1), $angle(l_1,l_2)=pi/2$, in example(2), $angle(l_1,l_2)=2pi/3$. What are all possible allowed angles between $l_1$ and $l_2$ (infinitely many or finitely many)? Or are $pi/2,2pi/3$ the only allowed values of $angle(l_1,l_2)$?
Here is an alternative formulation of the same problem: find all operators $Ain mathrm{End}(V^2)$ such that the set of operators ${A_{12}+C,A_{23}+C}$ in $mathrm{End}(V^3)$ generates the Lie algebra $mathfrak{su}(2)$, for a suitable $Cin mathrm{End}(V^3)$ that commutes with $A_{12},A_{23}$. In example (1) $A=sigma^zotimessigma^x,C=0$ and in example (2) $A=P/2,C=-(P_{12}+P_{23}+P_{13})/6$.
Any hints, references, or further examples will be welcomed. Thanks!
representation-theory lie-algebras
asked Nov 20 '18 at 18:28
Lagrenge
1208
add a comment |
Let $V=mathbb{C}^d$ be a $d$-dimensional vector space over $mathbb{C}$. The problem is to find all possible representations $rho:mathfrak{l}to mathrm{End}(V^3)$ of the Lie algebra $mathfrak{l}=mathfrak{su}(2)$ in the product space $V^3=Votimes Votimes V$ with the following property:
(*)There exists two elements $l_1,l_2in mathfrak{l}$ such that $rho(l_1)=A_{12}+Cequiv Aotimes mathbb{1}+C,rho(l_2)=A_{23}+Cequiv mathbb{1}otimes A +C$, where $Ain mathrm{End}(V^2)$ (i.e. acts in $Votimes V$), $mathbb{1}$ is the identity operator in $mathrm{End}(V)$, and $A_{12},A_{23}$ denotes two different ways of embedding of $A$ in $mathrm{End}(V^3)$. $Cin mathrm{End}(V^3)$ is a suitable "central element" that commutes with $rho(l)$ for $forall lin mathfrak{l}$.
Examples:
(1). For $d$=2, consider the representation $rho(s^x)=sigma^z_1 sigma^x_2/2,rho(s^z)=sigma^z_2 sigma^x_3/2,rho(s^y)=sigma^z_1 sigma^y_2sigma^x_3/2$, where $sigma^x,sigma^y,sigma^z$ are Pauli matrices. This is the 8-dimensional direct sum representation $frac{1}{2}oplusfrac{1}{2}oplusfrac{1}{2}oplusfrac{1}{2}$. We have $l_1=s^x,l_2=s^z,A=sigma^zotimessigma^x$ (or $A_{12}=sigma^z_1sigma^x_2,A_{23}=sigma^z_2sigma^x_3$), and $C=0$. So this representation satisfies the above property(*).
(2) For arbitrary $d$, consider the representation
begin{eqnarray}
rho(s^x)=frac{2P_{12}-P_{23}-P_{13}}{3sqrt{6}}+frac{M}{6i}\
rho(s^y)=frac{2P_{23}-P_{13}-P_{12}}{3sqrt{6}}+frac{M}{6i}\
rho(s^z)=frac{2P_{13}-P_{12}-P_{23}}{3sqrt{6}}+frac{M}{6i},
end{eqnarray}
where $Pinmathrm{End}(V^2)$ is the permutation operator defined by $P(e_1otimes e_2)=e_2otimes e_1$ for $forall e_1,e_2in V$, and $P_{12},P_{23},P_{13}$ are the different embedding of $P$ in $mathrm{End}(V^3)$, e.g. $P_{13}(e_1otimes e_2otimes e_3)=(e_3otimes e_2otimes e_1)$, etc., and $M=[P_{12},P_{23}]=P_{12}P_{23}-P_{23}P_{12}=[P_{23},P_{13}]=[P_{13},P_{12}]$. Put it more formally, $rho$ maps $mathfrak{su}(2)$ to the group algebra $mathbb{C}(S_3)$ of the symmetry group $S_3$. Therefore, let $l_1=frac{2s^x-s^y-s^z}{sqrt{6}},l_2=frac{2s^y-s^z-s^x}{sqrt{6}}$, we have $$rho(l_1)=frac{P_{12}}{2}-frac{c}{6},~~~rho(l_2)=frac{P_{23}}{2}-frac{c}{6},$$ where $c=P_{12}+P_{23}+P_{13}$ is a central element of $mathbb{C}(S_3)$ and therefore commutes with $rho(l)$ for $forall lin mathfrak{su}(2)$. Therefore, this representation satisfies the property(*) with $A=P/2$ and $C=-c/6$. When $d=2$, this representation is the 8-dimensional direct sum representation $0^4oplus frac{1}{2}oplusfrac{1}{2}$.
The above two are all the examples I know up to now. Is there a systematic way of finding all such representations? Interestingly, notice the angle between $l_1$ and $l_2$: in example(1), $angle(l_1,l_2)=pi/2$, in example(2), $angle(l_1,l_2)=2pi/3$. What are all possible allowed angles between $l_1$ and $l_2$ (infinitely many or finitely many)? Or are $pi/2,2pi/3$ the only allowed values of $angle(l_1,l_2)$?
Here is an alternative formulation of the same problem: find all operators $Ain mathrm{End}(V^2)$ such that the set of operators ${A_{12}+C,A_{23}+C}$ in $mathrm{End}(V^3)$ generates the Lie algebra $mathfrak{su}(2)$, for a suitable $Cin mathrm{End}(V^3)$ that commutes with $A_{12},A_{23}$. In example (1) $A=sigma^zotimessigma^x,C=0$ and in example (2) $A=P/2,C=-(P_{12}+P_{23}+P_{13})/6$.
Any hints, references, or further examples will be welcomed. Thanks!
representation-theory lie-algebras
asked Nov 20 '18 at 18:28
Lagrenge
1208
Let $V=mathbb{C}^d$ be a $d$-dimensional vector space over $mathbb{C}$. The problem is to find all possible representations $rho:mathfrak{l}to mathrm{End}(V^3)$ of the Lie algebra $mathfrak{l}=mathfrak{su}(2)$ in the product space $V^3=Votimes Votimes V$ with the following property:
(*)There exists two elements $l_1,l_2in mathfrak{l}$ such that $rho(l_1)=A_{12}+Cequiv Aotimes mathbb{1}+C,rho(l_2)=A_{23}+Cequiv mathbb{1}otimes A +C$, where $Ain mathrm{End}(V^2)$ (i.e. acts in $Votimes V$), $mathbb{1}$ is the identity operator in $mathrm{End}(V)$, and $A_{12},A_{23}$ denotes two different ways of embedding of $A$ in $mathrm{End}(V^3)$. $Cin mathrm{End}(V^3)$ is a suitable "central element" that commutes with $rho(l)$ for $forall lin mathfrak{l}$.
Examples:
(1). For $d$=2, consider the representation $rho(s^x)=sigma^z_1 sigma^x_2/2,rho(s^z)=sigma^z_2 sigma^x_3/2,rho(s^y)=sigma^z_1 sigma^y_2sigma^x_3/2$, where $sigma^x,sigma^y,sigma^z$ are Pauli matrices. This is the 8-dimensional direct sum representation $frac{1}{2}oplusfrac{1}{2}oplusfrac{1}{2}oplusfrac{1}{2}$. We have $l_1=s^x,l_2=s^z,A=sigma^zotimessigma^x$ (or $A_{12}=sigma^z_1sigma^x_2,A_{23}=sigma^z_2sigma^x_3$), and $C=0$. So this representation satisfies the above property(*).
(2) For arbitrary $d$, consider the representation
begin{eqnarray}
rho(s^x)=frac{2P_{12}-P_{23}-P_{13}}{3sqrt{6}}+frac{M}{6i}\
rho(s^y)=frac{2P_{23}-P_{13}-P_{12}}{3sqrt{6}}+frac{M}{6i}\
rho(s^z)=frac{2P_{13}-P_{12}-P_{23}}{3sqrt{6}}+frac{M}{6i},
end{eqnarray}
where $Pinmathrm{End}(V^2)$ is the permutation operator defined by $P(e_1otimes e_2)=e_2otimes e_1$ for $forall e_1,e_2in V$, and $P_{12},P_{23},P_{13}$ are the different embedding of $P$ in $mathrm{End}(V^3)$, e.g. $P_{13}(e_1otimes e_2otimes e_3)=(e_3otimes e_2otimes e_1)$, etc., and $M=[P_{12},P_{23}]=P_{12}P_{23}-P_{23}P_{12}=[P_{23},P_{13}]=[P_{13},P_{12}]$. Put it more formally, $rho$ maps $mathfrak{su}(2)$ to the group algebra $mathbb{C}(S_3)$ of the symmetry group $S_3$. Therefore, let $l_1=frac{2s^x-s^y-s^z}{sqrt{6}},l_2=frac{2s^y-s^z-s^x}{sqrt{6}}$, we have $$rho(l_1)=frac{P_{12}}{2}-frac{c}{6},~~~rho(l_2)=frac{P_{23}}{2}-frac{c}{6},$$ where $c=P_{12}+P_{23}+P_{13}$ is a central element of $mathbb{C}(S_3)$ and therefore commutes with $rho(l)$ for $forall lin mathfrak{su}(2)$. Therefore, this representation satisfies the property(*) with $A=P/2$ and $C=-c/6$. When $d=2$, this representation is the 8-dimensional direct sum representation $0^4oplus frac{1}{2}oplusfrac{1}{2}$.
The above two are all the examples I know up to now. Is there a systematic way of finding all such representations? Interestingly, notice the angle between $l_1$ and $l_2$: in example(1), $angle(l_1,l_2)=pi/2$, in example(2), $angle(l_1,l_2)=2pi/3$. What are all possible allowed angles between $l_1$ and $l_2$ (infinitely many or finitely many)? Or are $pi/2,2pi/3$ the only allowed values of $angle(l_1,l_2)$?
Here is an alternative formulation of the same problem: find all operators $Ain mathrm{End}(V^2)$ such that the set of operators ${A_{12}+C,A_{23}+C}$ in $mathrm{End}(V^3)$ generates the Lie algebra $mathfrak{su}(2)$, for a suitable $Cin mathrm{End}(V^3)$ that commutes with $A_{12},A_{23}$. In example (1) $A=sigma^zotimessigma^x,C=0$ and in example (2) $A=P/2,C=-(P_{12}+P_{23}+P_{13})/6$.
Any hints, references, or further examples will be welcomed. Thanks!
representation-theory lie-algebras
representation-theory lie-algebras
asked Nov 20 '18 at 18:28
Lagrenge
1208
asked Nov 20 '18 at 18:28
Lagrenge
1208
asked Nov 20 '18 at 18:28
Lagrenge
1208
asked Nov 20 '18 at 18:28
Lagrenge
1208
asked Nov 20 '18 at 18:28
Lagrenge
1208
1208
add a comment |
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