Mixing
Real roots of U(2)
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On page 350 of Hall's book real roots of $U(2)$ are listed as $(1, 1)$ and $(-1, -1)$ after identifying the maximal torus algebra $frak{t}$ of diagonal matrices with $mathbb{R}^2$. However, my calculations show that the roots should be $(1, -1)$ and $(-1, 1)$, and I don't see what am I missing. Here are the calculations:
Let $H=text{diag}(ai, bi)$ and $$X_1=begin{bmatrix}0&1\-1&0end{bmatrix}, X_2=begin{bmatrix}0&i\i&0end{bmatrix}.$$
Then $[H, X_1]=(a-b)X_2$ and $[H, X_2]=(b-a)X_1$ which implies that
$$[H, X_1+iX_2]=-i(a-b)(X_1+iX_2), quad [H, X_1-iX_2]=i(a-b)(X_1-iX_2).$$ Thus, the real root corresponding to $X+iX_2in mathfrak{t}_mathbb{C}$ is $alpha=text{diag}(-i, i)in mathfrak{t}$ because
$$langle alpha , Hrangle =text{tr}(alpha^*H)=b-a$$.
I suspect that we are using different inner products (because at the bottom of page 351 apparently the author gets $langle alpha, alpharangle=1$.) Any clarification is appreciated.
lie-algebras root-systems
asked Jan 28 at 2:24
SimonSimon
289111
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add a comment |
$begingroup$
On page 350 of Hall's book real roots of $U(2)$ are listed as $(1, 1)$ and $(-1, -1)$ after identifying the maximal torus algebra $frak{t}$ of diagonal matrices with $mathbb{R}^2$. However, my calculations show that the roots should be $(1, -1)$ and $(-1, 1)$, and I don't see what am I missing. Here are the calculations:
Let $H=text{diag}(ai, bi)$ and $$X_1=begin{bmatrix}0&1\-1&0end{bmatrix}, X_2=begin{bmatrix}0&i\i&0end{bmatrix}.$$
Then $[H, X_1]=(a-b)X_2$ and $[H, X_2]=(b-a)X_1$ which implies that
$$[H, X_1+iX_2]=-i(a-b)(X_1+iX_2), quad [H, X_1-iX_2]=i(a-b)(X_1-iX_2).$$ Thus, the real root corresponding to $X+iX_2in mathfrak{t}_mathbb{C}$ is $alpha=text{diag}(-i, i)in mathfrak{t}$ because
$$langle alpha , Hrangle =text{tr}(alpha^*H)=b-a$$.
I suspect that we are using different inner products (because at the bottom of page 351 apparently the author gets $langle alpha, alpharangle=1$.) Any clarification is appreciated.
lie-algebras root-systems
asked Jan 28 at 2:24
SimonSimon
289111
$endgroup$
add a comment |
$begingroup$
On page 350 of Hall's book real roots of $U(2)$ are listed as $(1, 1)$ and $(-1, -1)$ after identifying the maximal torus algebra $frak{t}$ of diagonal matrices with $mathbb{R}^2$. However, my calculations show that the roots should be $(1, -1)$ and $(-1, 1)$, and I don't see what am I missing. Here are the calculations:
Let $H=text{diag}(ai, bi)$ and $$X_1=begin{bmatrix}0&1\-1&0end{bmatrix}, X_2=begin{bmatrix}0&i\i&0end{bmatrix}.$$
Then $[H, X_1]=(a-b)X_2$ and $[H, X_2]=(b-a)X_1$ which implies that
$$[H, X_1+iX_2]=-i(a-b)(X_1+iX_2), quad [H, X_1-iX_2]=i(a-b)(X_1-iX_2).$$ Thus, the real root corresponding to $X+iX_2in mathfrak{t}_mathbb{C}$ is $alpha=text{diag}(-i, i)in mathfrak{t}$ because
$$langle alpha , Hrangle =text{tr}(alpha^*H)=b-a$$.
I suspect that we are using different inner products (because at the bottom of page 351 apparently the author gets $langle alpha, alpharangle=1$.) Any clarification is appreciated.
lie-algebras root-systems
asked Jan 28 at 2:24
SimonSimon
289111
$endgroup$
On page 350 of Hall's book real roots of $U(2)$ are listed as $(1, 1)$ and $(-1, -1)$ after identifying the maximal torus algebra $frak{t}$ of diagonal matrices with $mathbb{R}^2$. However, my calculations show that the roots should be $(1, -1)$ and $(-1, 1)$, and I don't see what am I missing. Here are the calculations:
Let $H=text{diag}(ai, bi)$ and $$X_1=begin{bmatrix}0&1\-1&0end{bmatrix}, X_2=begin{bmatrix}0&i\i&0end{bmatrix}.$$
Then $[H, X_1]=(a-b)X_2$ and $[H, X_2]=(b-a)X_1$ which implies that
$$[H, X_1+iX_2]=-i(a-b)(X_1+iX_2), quad [H, X_1-iX_2]=i(a-b)(X_1-iX_2).$$ Thus, the real root corresponding to $X+iX_2in mathfrak{t}_mathbb{C}$ is $alpha=text{diag}(-i, i)in mathfrak{t}$ because
$$langle alpha , Hrangle =text{tr}(alpha^*H)=b-a$$.
I suspect that we are using different inner products (because at the bottom of page 351 apparently the author gets $langle alpha, alpharangle=1$.) Any clarification is appreciated.
lie-algebras root-systems
lie-algebras root-systems
asked Jan 28 at 2:24
SimonSimon
289111
asked Jan 28 at 2:24
SimonSimon
289111
asked Jan 28 at 2:24
SimonSimon
289111
asked Jan 28 at 2:24
SimonSimon
289111
asked Jan 28 at 2:24
SimonSimon
289111
289111
add a comment |
add a comment |
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