A About computing the tangent space at 1 of certain lie groups

aalma
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Hello :),

I am wondering of the right and direct method to calculate the following tangent spaces at ##1##: ##T_ISL_n(R)##, ##T_IU(n)## and ##T_ISU(n)##.

Definitions I know:
Given a smooth curve ##γ : (− ,) → R^n## with ##γ(0) = x##, a tangent vector ##˙γ(0)## is a vector with components ##(˙γ1,..., ˙γn)##.
Two (smooth) curves ##γ,γ': (− ,) → M## with ##γ(0) = γ'(0) = x## are called equivalent if in a certain chart containing x (or in any chart) they define the same tangent vector at ##x##.
We define ##T_x(M)## to be the set of equivalence classes of curves as above. We note that ##T_x(M)## is a vector space: Choose a chart ##a : U → R^n## containing x. Then the assignment ##γ → d/dt (a ◦ γ)(0)## defines a bijection from ##T_x(M)## to ##R^n##. Different charts define different bijections, but they differ by the Jacobi matrix: if ##b : U → R^n## is another chart, the bijections from ##Tx(M)## to ##R^n## differ by the Jacobi matrix at ##a(x)## of the transformation ##b◦a^{−1} : a(U) → b(U)##.
Example: Let ##G = SO(n,R)## and let ##γ## be a curve in ##G## with ##γ(0) = 1##. Then we can write ##γ(s) = 1 + sA + o(s^2)## where ##A ∈ M_{n× n}## and we want to describe possible values for ##A##. Obviously ##γ(s)^t = 1 + sA^t + o(s^2)## so ##1 = γ(s)γ(s)^t = 1 + s(A + A^t) + o(s^2)## which implies that ##A## is skewsymmetric. In the opposite direction, if ##A## is skew-symmetric, ##γ(s) = e^{sA}## is orthogonal and ##e^{sA} = 1 + sA + o(s^2)##. We have proven that ##T_1(G)## in this case is the vector space of skew-symmetric matrices.

Can you please suggest a way to compute such tangent spaces.
 
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aalma said:
Is there a way to calculate the above tangent spaces in a way similar to the example I added?
You need to define curves ##\gamma \, : \,(-1,1) \longrightarrow M## with ##\gamma(0)=x## where ##x=I## is the identity matrix in your case. Then you expand ##\gamma(t)## into a Taylor series, and ##\dot\gamma(0)## will be your tangent vectors. Next, gather so many curves ##\gamma(t) ## such that you cover all ##\operatorname{dim}M## many possible directions to be able to span ##T_x(M).##

The procedure in the link does the same, only that it uses the canonical Cartesian coordinate system:
$$
T_x(M)=\operatorname{span}\left\{\sum_{i,j}^{\operatorname{dim}M} \left. a_{ij} \dfrac{\partial }{\partial x_{ij}}\right|_{t=0} \gamma(t)\right\}
$$
and the obvious curves
\begin{align*}
\gamma \, : \,(-1,1) &\longrightarrow M\\
t &\longmapsto \begin{pmatrix}1 &0 &\ldots &0\\ \ldots &x_{ij}&\ldots&0 \\ 0&0&\ldots& 1\end{pmatrix} \in M
\end{align*}
plus the defining equations, e.g. ##\det(a_{ij})=1## and / or ##(a_{ij}) + (a_{ij})^*=0.##The next problem is to find all curves. These are ##\operatorname{dim}M## many different ones since ##\operatorname{dim}T_x(M)=\operatorname{dim}M.##

In the linked article, I have chosen the easiest ones, still given the Cartesian coordinate system. E.g. a typical element of ##M=\operatorname{SL}(2)## looks like
\begin{align*}
X(t)&=\begin{pmatrix}x_{11}(t)&x_{12}(t)\\ x_{21}(t)&x_{22}(t) \end{pmatrix}\; \\&\wedge \;x_{11}(t)\cdot x_{22}(t)- x_{21}(t)\cdot x_{12}(t)=1\; \\&\wedge \;x_{11}(0)=1\, , \,x_{12}(0)=0\, , \,x_{21}(0)=0\, , \,x_{22}(0)=1
\end{align*}

and a typical curve accordingly
$$
\gamma (t)=X(t)\, , \,\gamma(0)=X(0)=I
$$
Differentiation for those matrices is easier than the full Taylor expansion. We are only interested in the second term evaluated at the identity matrix ##I## anyway. This would be the calculation in equations (26) and (27) for the determinant condition:
$$
\left. \dfrac{d}{dt}\right|_{t=0}\operatorname{det}(\gamma(t))=\left. \dfrac{d}{dt}\right|_{t=0}1=0=\operatorname{trace}\left(a_{ij}\right)
$$

Of course, you can choose your own charts (providing the coordinate system), curves (providing the tangent vectors at ##x=I##), and calculate the entire Taylor series, but the Cartesian charts, coordinate functions ##t\mapsto (x_{ij}(t))##, and the first derivative ##\left(\left. \dfrac{d}{dt}\right|_{t=0}x_{ij}(t)\right)## are likely the easiest you can get. The crucial condition is ##\gamma(t)\in M## with ##\gamma(0)=I.##
 
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fresh_42 said:
You need to define curves ##\gamma \, : \,(-1,1) \longrightarrow M## with ##\gamma(0)=x## where ##x=I## is the identity matrix in your case. Then you expand ##\gamma(t)## into a Taylor series, and ##\dot\gamma(0)## will be your tangent vectors. Next, gather so many curves ##\gamma(t) ## such that you cover all ##\operatorname{dim}M## many possible directions to be able to span ##T_x(M).##

The procedure in the link does the same, only that it uses the canonical Cartesian coordinate system:
$$
T_x(M)=\operatorname{span}\left\{\sum_{i,j}^{\operatorname{dim}M} \left. a_{ij} \dfrac{\partial }{\partial x_{ij}}\right|_{t=0} \gamma(t)\right\}
$$
and the obvious curves
\begin{align*}
\gamma \, : \,(-1,1) &\longrightarrow M\\
t &\longmapsto \begin{pmatrix}1 &0 &\ldots &0\\ \ldots &x_{ij}&\ldots&0 \\ 0&0&\ldots& 1\end{pmatrix} \in M
\end{align*}
plus the defining equations, e.g. ##\det(a_{ij})=1## and / or ##(a_{ij}) + (a_{ij})^*=0.##The next problem is to find all curves. These are ##\operatorname{dim}M## many different ones since ##\operatorname{dim}T_x(M)=\operatorname{dim}M.##

In the linked article, I have chosen the easiest ones, still given the Cartesian coordinate system. E.g. a typical element of ##M=\operatorname{SL}(2)## looks like
\begin{align*}
X(t)&=\begin{pmatrix}x_{11}(t)&x_{12}(t)\\ x_{21}(t)&x_{22}(t) \end{pmatrix}\; \\&\wedge \;x_{11}(t)\cdot x_{22}(t)- x_{21}(t)\cdot x_{12}(t)=1\; \\&\wedge \;x_{11}(0)=1\, , \,x_{12}(0)=0\, , \,x_{21}(0)=0\, , \,x_{22}(0)=1
\end{align*}

and a typical curve accordingly
$$
\gamma (t)=X(t)\, , \,\gamma(0)=X(0)=I
$$
Differentiation for those matrices is easier than the full Taylor expansion. We are only interested in the second term evaluated at the identity matrix ##I## anyway. This would be the calculation in equations (26) and (27) for the determinant condition:
$$
\left. \dfrac{d}{dt}\right|_{t=0}\operatorname{det}(\gamma(t))=\left. \dfrac{d}{dt}\right|_{t=0}1=0=\operatorname{trace}\left(a_{ij}\right)
$$

Of course, you can choose your own charts (providing the coordinate system), curves (providing the tangent vectors at ##x=I##), and calculate the entire Taylor series, but the Cartesian charts, coordinate functions ##t\mapsto (x_{ij}(t))##, and the first derivative ##\left(\left. \dfrac{d}{dt}\right|_{t=0}x_{ij}(t)\right)## are likely the easiest you can get. The crucial condition is ##\gamma(t)\in M## with ##\gamma(0)=I.##
fresh_42 said:
You need to define curves ##\gamma \, : \,(-1,1) \longrightarrow M## with ##\gamma(0)=x## where ##x=I## is the identity matrix in your case. Then you expand ##\gamma(t)## into a Taylor series, and ##\dot\gamma(0)## will be your tangent vectors. Next, gather so many curves ##\gamma(t) ## such that you cover all ##\operatorname{dim}M## many possible directions to be able to span ##T_x(M).##

The procedure in the link does the same, only that it uses the canonical Cartesian coordinate system:
$$
T_x(M)=\operatorname{span}\left\{\sum_{i,j}^{\operatorname{dim}M} \left. a_{ij} \dfrac{\partial }{\partial x_{ij}}\right|_{t=0} \gamma(t)\right\}
$$
and the obvious curves
\begin{align*}
\gamma \, : \,(-1,1) &\longrightarrow M\\
t &\longmapsto \begin{pmatrix}1 &0 &\ldots &0\\ \ldots &x_{ij}&\ldots&0 \\ 0&0&\ldots& 1\end{pmatrix} \in M
\end{align*}
plus the defining equations, e.g. ##\det(a_{ij})=1## and / or ##(a_{ij}) + (a_{ij})^*=0.##The next problem is to find all curves. These are ##\operatorname{dim}M## many different ones since ##\operatorname{dim}T_x(M)=\operatorname{dim}M.##

In the linked article, I have chosen the easiest ones, still given the Cartesian coordinate system. E.g. a typical element of ##M=\operatorname{SL}(2)## looks like
\begin{align*}
X(t)&=\begin{pmatrix}x_{11}(t)&x_{12}(t)\\ x_{21}(t)&x_{22}(t) \end{pmatrix}\; \\&\wedge \;x_{11}(t)\cdot x_{22}(t)- x_{21}(t)\cdot x_{12}(t)=1\; \\&\wedge \;x_{11}(0)=1\, , \,x_{12}(0)=0\, , \,x_{21}(0)=0\, , \,x_{22}(0)=1
\end{align*}

and a typical curve accordingly
$$
\gamma (t)=X(t)\, , \,\gamma(0)=X(0)=I
$$
Differentiation for those matrices is easier than the full Taylor expansion. We are only interested in the second term evaluated at the identity matrix ##I## anyway. This would be the calculation in equations (26) and (27) for the determinant condition:
$$
\left. \dfrac{d}{dt}\right|_{t=0}\operatorname{det}(\gamma(t))=\left. \dfrac{d}{dt}\right|_{t=0}1=0=\operatorname{trace}\left(a_{ij}\right)
$$

Of course, you can choose your own charts (providing the coordinate system), curves (providing the tangent vectors at ##x=I##), and calculate the entire Taylor series, but the Cartesian charts, coordinate functions ##t\mapsto (x_{ij}(t))##, and the first derivative ##\left(\left. \dfrac{d}{dt}\right|_{t=0}x_{ij}(t)\right)## are likely the easiest you can get. The crucial condition is ##\gamma(t)\in M## with ##\gamma(0)=I.##
Thanks. Can you please give more deatils of how to do the calculations with taylor series expansion?
 

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