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BookInquiry
 Post subject: How to solve: AX+XB=C, X=? I tried many times but fail...
PostPosted: Mon, 20 Aug 2012 09:40:40 UTC 
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How to solve: AX+XB=C, X=?
It seems not too difficult, I tried many times but fail
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outermeasure
 Post subject: Re: How to solve: AX+XB=C, X=? I tried many times but fail..
PostPosted: Mon, 20 Aug 2012 09:51:33 UTC 
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BookInquiry wrote:
How to solve: AX+XB=C, X=?
It seems not too difficult, I tried many times but fail


X\mapsto AX+XB is an \mathbb{F}-linear map on M_{n\times n}(\mathbb{F}), which may or may not be surjective depending on the particular A,B\in M_{n\times n}(\mathbb{F}).

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\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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