Lie Groups and Lie Algebras: Winter-2023 HW 1 : Due Wednesday 1/18
Bilinear forms
S0 4604S
Suppose \(G\) is an \(n\times n\) matrix. Define a “dot product” on \({\mathbb R}^n\) by
\[
\boldsymbol{\vec v}\cdot\boldsymbol{\vec w} = \boldsymbol{\vec v}^T G \boldsymbol{\vec w}\
\]
What conditions on \(G\) guarantee that this dot product is a symmetric,
non-degenerate bilinear form on \({\mathbb R}^n\)?
Are your conditions necessary as well as sufficient?
Suppose \(T:{\mathbb R}^n\to{\mathbb R}^n\) is a linear transformation with matrix \(M\).
What condition on \(M\) is equivalent to \(T(\boldsymbol{\vec v})\cdot T(\boldsymbol{\vec w})=\boldsymbol{\vec v}\cdot\boldsymbol{\vec w}\) for all \(\boldsymbol{\vec v},\boldsymbol{\vec w}\in{\mathbb R}^n\)?
Suppose \(G\) is an \(n\times n\) matrix. Define a “dot product” on \({\mathbb R}^n\) by
\[
\boldsymbol{\vec v}\cdot\boldsymbol{\vec w} = \boldsymbol{\vec v}^T G \boldsymbol{\vec w}
\]
What conditions on \(G\) guarantee that this dot product is an anti-symmetric,
non-degenerate bilinear form on \({\mathbb R}^n\)?
Are your conditions necessary as well as sufficient?
Show that your conditions imply that \(n\) is even.
Vector Fields on $\mathbb{S}^3$
S0 4604S
Show that the vector fields
\begin{align*}
r_z &= \partial_\phi ,\\
r_x &=
-\sin\phi \,\partial_\theta - \cot\theta\cos\phi \,\partial_\phi
+ \csc\theta\cos\phi \,\partial_\psi ,\\
r_y &=
\cos\phi \,\partial_\theta - \cot\theta\sin\phi \,\partial_\phi
+ \csc\theta\sin\phi \,\partial_\psi
\end{align*}
satisfy the expected commutation relations, that is,
\begin{equation*}
[r_x,r_y] = -r_z ,
\qquad [r_y,r_z] = -r_x ,
\qquad [r_z,r_x] = -r_y .
\end{equation*}