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Jianrong Li
Jianrong Li
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For the Lie algebra $\mathfrak{so}(2n+1, \mathbb{C})$, there is a matrix representation given by the following matrices: \begin{align} \left( \begin{matrix} 0 & x & y \\ -y^T & A & B \\ -x^T & C & -A^T \end{matrix} \right), \end{align} where $x, y$ are $1 \times n$ matrices, $A,B,C$ are $n \times n$ matrices. Are there similar matrix representations for Lie group $\mathrm{SO}(2n+1, \mathbb{C})$? Thank you very much.

Edit: I forgot to mention that $B,C$ are skew-symmetric, $B=-B^T, C=-C^T$.

For the Lie algebra $\mathfrak{so}(2n+1, \mathbb{C})$, there is a matrix representation given by the following matrices: \begin{align} \left( \begin{matrix} 0 & x & y \\ -y^T & A & B \\ -x^T & C & -A^T \end{matrix} \right), \end{align} where $x, y$ are $1 \times n$ matrices, $A,B,C$ are $n \times n$ matrices. Are there similar matrix representations for Lie group $\mathrm{SO}(2n+1, \mathbb{C})$? Thank you very much.

For the Lie algebra $\mathfrak{so}(2n+1, \mathbb{C})$, there is a matrix representation given by the following matrices: \begin{align} \left( \begin{matrix} 0 & x & y \\ -y^T & A & B \\ -x^T & C & -A^T \end{matrix} \right), \end{align} where $x, y$ are $1 \times n$ matrices, $A,B,C$ are $n \times n$ matrices. Are there similar matrix representations for Lie group $\mathrm{SO}(2n+1, \mathbb{C})$? Thank you very much.

Edit: I forgot to mention that $B,C$ are skew-symmetric, $B=-B^T, C=-C^T$.

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YCor
YCor
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Reference request: matrix Matrix representations of Lie groups of type $B_n$

For the Lie algebra $so(2n+1, \mathbb{C})$. There$\mathfrak{so}(2n+1, \mathbb{C})$, there is a matrix representation given by the following matrices: \begin{align} \left( \begin{matrix} 0 & x & y \\ -y^T & A & B \\ -x^T & C & -A^T \end{matrix} \right), \end{align} where $x, y$ are $1 \times n$ matrices, $A,B,C$ are $n \times n$ matrices. Are there similar matrix representations for Lie group $SO(2n+1, \mathbb{C})$$\mathrm{SO}(2n+1, \mathbb{C})$? Thank you very much.

Reference request: matrix representations of Lie groups of type $B_n$

For Lie algebra $so(2n+1, \mathbb{C})$. There is a matrix representation given by the following matrices: \begin{align} \left( \begin{matrix} 0 & x & y \\ -y^T & A & B \\ -x^T & C & -A^T \end{matrix} \right), \end{align} where $x, y$ are $1 \times n$ matrices, $A,B,C$ are $n \times n$ matrices. Are there similar matrix representations for Lie group $SO(2n+1, \mathbb{C})$? Thank you very much.

Matrix representations of Lie groups of type $B_n$

For the Lie algebra $\mathfrak{so}(2n+1, \mathbb{C})$, there is a matrix representation given by the following matrices: \begin{align} \left( \begin{matrix} 0 & x & y \\ -y^T & A & B \\ -x^T & C & -A^T \end{matrix} \right), \end{align} where $x, y$ are $1 \times n$ matrices, $A,B,C$ are $n \times n$ matrices. Are there similar matrix representations for Lie group $\mathrm{SO}(2n+1, \mathbb{C})$? Thank you very much.

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Jianrong Li
Jianrong Li
  • 6.2k
  • 2
  • 21
  • 34

Reference request: matrix representations of Lie groups of type $B_n$

For Lie algebra $so(2n+1, \mathbb{C})$. There is a matrix representation given by the following matrices: \begin{align} \left( \begin{matrix} 0 & x & y \\ -y^T & A & B \\ -x^T & C & -A^T \end{matrix} \right), \end{align} where $x, y$ are $1 \times n$ matrices, $A,B,C$ are $n \times n$ matrices. Are there similar matrix representations for Lie group $SO(2n+1, \mathbb{C})$? Thank you very much.