## Lie groups and representations

A (complex) **Lie group** is a finite dimensional smooth (complex) manifold $G$ together with a group structure on $G$, such that the multiplication $G \times G \rightarrow G$ and the attaching of an inverse $g \mapsto g^1:G \rightarrow G$ are smooth maps.

A **matrix Lie** group is any subgroup $H$ of $GL(n;C)$. $H$ is a closed subset: if $A_n$ is any sequence of matrices in $H$, and $A_n$ converges to some matrix $A$, then either $A \in H$, or $A$ is not invertible.

### Counterexamples.  
Set of all $n \times n$ invertible matrices all of whose entries are rational. This is in fact a subgroup of $GL(n;\mathbb{C})$, but not a closed subgroup.  
Subgroup of $GL(n;\mathbb{C})$ : Let $a$ be an irrational real number, and let
$H = 
\begin{Bmatrix}
\begin{pmatrix}
e^{it} & 0\\ 
0 & e^{ita}
\end{pmatrix}
\vert t \in \mathbb{R}
\end{Bmatrix}
$

We can find a sequence wich converges to $-I$, but it does not belong to $H$. The closure of H is a 2 dimensional torus: $H = 
\begin{Bmatrix}
\begin{pmatrix}
e^{it} & 0\\ 
0 & e^{is}
\end{pmatrix}
\vert t,s \in \mathbb{R}
\end{Bmatrix}
$


### Examples.
General linear groups: $GL(n;\mathbb{R}) \lt GL(n;\mathbb{C})$

Special linear groups: $SL(n;\mathbb{R})$ and $SL(n;\mathbb{C})$, matrices having determinant one.

orthogonal and special orthogonal groups: $O(n), SO(n)$. real matrices  
> column vectors that make up $A$ are orthonormal, that is, $\sum_{i=1}^nA_{ij}A_{ik} = \delta_{jk}$  
> $\langle x,y \rangle = \langle Ax,Ay \rangle$  
> $A^{tr} A = I$  
$det(A) = \pm 1$

unitary and special unitary groups, $U(n)$ and $SU(n)$: $A^\dagger A = I$  
$\vert det(A) \vert = 1$

complex orthogonal groups, $O(n;\mathbb{C})$ and $SO(n;\mathbb{C})$

generalized orthogonal $O(n;k)$  
> Define a symmetric bilinear form $[...]_{n+k}$ on $\mathbb{R}^{n+k}$: $$[x,y]_{n,k} = x_1 y_1 + ... + x_n y_n - x_{n+1} y_{n+1} - ... - x_{n+k} y_{n+k}$$    
> $A \in O(n;k) \Longrightarrow [Ax,Ay]=[x,y] for all x,y \in \mathbb{R}^{n+k}$  
> $A^{tr} g A = g$ where $g$ is diagonal matrix (metric tensor) with 1 in first $n$ positions and -1 in last $k$.  
$det(A) = \pm 1$

Lorentz group: $O(3; 1)$

symplectic groups $Sp(n;\mathbb{R})$, $Sp(n;\mathbb{C})$, and $Sp(n)$  
> Consider the skew-symmetric bilinear form $B$ on $\mathbb{R}^{2n}$ defined as: $B[x,y] = \sum_{i=1}^n x_i y_{n+i} - x_{n+i} y_i$  
> $B[Ax,Ay] = B[x,y]$ for all $x,y$  
> if $J(2n \times 2n) = 
\begin{pmatrix}
0 & I\\ 
-I & 0
\end{pmatrix}$, then $A^{tr} J A = J$  
> $Sp(n) = Sp(n;\mathbb{C}) \cap U(2n)$  
> $det(A) = 1$

The Heisenberg group $H$:  $A = 
\begin{pmatrix}
1 & a & b\\ 
0 & 1 & c\\ 
0 & 0 & 1
\end{pmatrix}
$; $A^{-1} = 
\begin{pmatrix}
1 & -a & ac-b\\ 
0 & 1 & -c\\ 
0 & 0 & 1
\end{pmatrix}
$; 
$H$ is subgroup of GL(n;\mathbb{R})$


$\mathbb{R}^* \simeq GL(1;\mathbb{R})$, $\mathbb{C}^* \simeq GL(1;\mathbb{C})$, $S^1 \simeq U(1)$, 
$\mathbb{R}^n \simeq \begin{pmatrix}
e^{x_1} &   & 0\\ 
  & ... &  \\ 
0 &   & e^{x_n}
\end{pmatrix}$



### Euclidean and Poincar´e groups.

Euclidean group $E(n)$ is by definition the group of all one-to-one, onto, distance-preserving maps of $\mathbb{R}^n$
to itself, that is, maps $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $d (f (x) , f (y)) = d (x, y)$ for all
$x, y \in \mathbb{R}^n$.$E(n)$ is subgroup of $GL(n+1;\mathbb{R})$. $T = T_x R$ where $R \in O(n)$ and $T_x$ is translation by $x$  
$T = \begin{pmatrix}
 &  &  & x_1\\ 
 & R &  & \vdots\\ 
 &  &  & x_n\\ 
0 & ...  & 0 & 1
\end{pmatrix}$

Poincar´e group P(n; 1). affine transformations of $\mathbb{R}^{n+1}$ which preserve the Lorentz distance $d_L(x, y) = (x_1 − y_1)^2 + · · · + (x_n − y_n)^2 − (x_{n+1} − y_{n+1})^2$. $T = T_x R$ where $R \in O(n;1)$

A matrix Lie group $G$ is said to be **compact** if it is closed ($A_n \leftarrow A \in G$) and bounded ($A_{ij} \lt Constant$)

A matrix Lie group $G$ is said to be **connected** if given any two matrices $A$ and $B$ in $G$, there exists a continuous path $A(t), a \le t \le b$, lying in $G$ with $A(a) = A$, and $A(b) = B$.  
For matrix Lie groups topological definitions of path-connected and connected are identical.

If $G$ is a matrix Lie group, then the component of $G$ containing the identity is a subgroup of $G$.

A connected matrix Lie group $G$ is said to be simply connected if every loop in $G$ can be shrunk continuously to a point in $G$.  
More precisely, $G$ is **simply connected** if given any continuous path $A(t), 0 ≤ t ≤ 1$, lying in $G$ with $A(0) = A(1)$, there exists a continuous function $A(s, t), 0 \le s, t \le 1$, taking values in $G$ with the following properties:  
>1) $A(s, 0) = A(s, 1)$ for all $s$  
>2) $A(0, t) = A(t)$  
>3) $A(1, t) = A(1, 0)$ for all $t$



$$\begin{matrix}
 & Components & Simply \space connected & dim & \\ 
GL(n;\mathbb{C}) & 1 & no & n^2 & \\ 
SL(n;\mathbb{C}) & 1  & yes & n^2 & \\
GL(n;\mathbb{R}) & 2 & no & n^2 & \\ 
SL(n;\mathbb{R}) & 1 & no & n^2-1 & \\ 
O(n) & 2 &  & \frac{n (n-1)}{2} & \\ 
SO(n) & 1 & no & \frac{n (n-1)}{2} & \\ 
U(n) & 1 & no & n^2 & \\ 
SU(n) & 1 & yes & n^2-1 & \\ 
O(n;1) & 4 &  &  & \\ 
SO(n;1) & 2 & n=1 yes; n \gt 1 no &  & \\ 
Heisenberg & 1 & yes &  & \\ 
E(n) & 2 &  &  & \\ 
P(n;1) & 4 &  &  & \\ 
 &  &  &  & 
\end{matrix}$$

Theorem: Every matrix Lie group is a Lie group.

Theorem: Let $G$ and $H$ be Lie groups, and $φ$ a group homomorphism from $G$ to $H$. Then if $φ$ is continuous it is also smooth.

## Lie Algebras and the Exponential Mapping

For any $n \times n$ real or complex matrix $X$, the series $e^X = \sum_{m=0}^\infty \frac{X^m}{m!}$ converges. The matrix exponential $e^X$ is a continuous function of $X$.

$\Vert A \Vert = sup_{x \neq 0} \frac{\Vert Ax \Vert}{\Vert x \Vert}$ $\Leftrightarrow$ $\Vert A \Vert$ is the smallest $\lambda$, $\Vert Ax \Vert \le \lambda \Vert x \Vert$ for $x \in $\mathbb{C}^n$

$\sum_{m=0}^\infty \Vert \frac{X^m}{m!} \Vert \le \sum_{m=0}^\infty \frac{\Vert X \Vert ^m}{m!} = e^{\Vert X \Vert}$. (The sum converges absolutely) 

Let $X, Y$ be arbitrary $n \times n$ matrices. Then
> $e^0 = I$  
> $(e^X)^{-1} = e^{-X}$  
> $e^{(\alpha + \beta)X} = e^{\alpha X} e^{\beta X}$  
> if $XY = YX$, then $e^{X+Y} = e^X e^Y = e^y e^x$  
> $e^{C X C^{-1}} = C e^X C^{-1}$  
> $\Vert e^X \Vert \le e^{\Vert X \Vert}$

$e^{tX}$ is a smooth curve in $\mathbb{C}^{n \times n}$. And $\frac{d}{dt} e^{tX} = X e^{tX} = e^{tX} X$.  
And particularly $\left . \frac{d}{dt} \right |_{t=0} e^{tX} = X e^{tX} = e^{tX} X$.

A Jordan–Chevalley decomposition of $X$ is an expression of it as a sum:  
$X = X_{ss} + X_n$, where $X_{ss}$ is semisimple, $X_n$ is nilpotent, and $X_{ss}$ and $X_n$ commute. If such a decomposition exists it is unique, and $X_{ss}$ and $X_n$ are in fact expressible as polynomials in $X$.

The function $log(z) = \sum_{m=1}^\infty (-1)^{m+1} \frac{(z-1)^m}{m}$ is defined and analytic in a circle of radius one about $z=1$.  
for all $\vert z-1 \vert \lt 1$ $e^{logz} = z$  
for all $\vert u \vert \lt log2$, $\vert e^u-1 \vert \lt 1$ and $log e^u = u$.

The function $log(A) = \sum_{m=1}^\infty (-1)^{m+1} \frac{(A-I)^m}{m}$ is defined and continuous on the set of all $n \times n$ complex matrices $A$ with $\Vert A-I \Vert \lt 1$, and $log A$ is real if $A$ is real.  
for all $\Vert A-I \Vert \lt 1$ $e^{logA} = A$  
for all $\Vert X \Vert \lt log2$, $\Vert e^A-1 \Vert \lt 1$ and $log e^X = X$.

Lie Product Formula (Trotter formula): $e^{X+Y} = \lim_{m \to \infty} (e^{\frac{X}{m}}e^{\frac{Y}{m}})^m$

Theorem: $det(e^X)=e^{tr(X)}$

$A : \mathbb{R} \to GL(n;\mathbb{C})$ is called a one-parameter group if  
> $A$ is continuous  
> $A(0) = I$  
> $A(t+s) = A(t) A(s)$ for all $t,s \in \mathbb{R}$

If $A$ is a one-parameter group in $GL(n;\mathbb{C})$, then there exists unique $X$, such that $A(t) = e^{tX}$



Let $G$ be a matrix Lie group. Then the **Lie algebra** of $G$, denoted $\mathfrak{g}$, is the set of all matrices $X$ such that $e^{tX}$ is in $G$ for all real numbers $t$. (In physics usually $e^{itX}$ is used).

$GL(n;\mathbb{C}) \Longrightarrow gl(n;\mathbb{C})$, all complex matrices.  
$GL(n;\mathbb{R}) \Longrightarrow gl(n;\mathbb{g})$, all real matrices.  
$SL(n;\mathbb{C}) \Longrightarrow sl(n;\mathbb{C})$, all complex matrices with trace 0.  
$SL(n;\mathbb{R}) \Longrightarrow sl(n;\mathbb{R})$, all real matrices with trace 0.  
$U(n) \Longrightarrow u(n)$, all complex matrices that $X^\dagger = -X$  
$su(n)$, complex matrices that $X^\dagger = -X$ and $tr(X)=0$  
$O(n)$ and $SO(n) \Longrightarrow so(n)$, all real such that $X^{tr} = -X$  
$o(n;k), so(n;k)$ all real such that $gX^{tr}g = X$  
$sp(n;\mathbb{R})$ all that $J X^{tr} J = X$  
Heisenberg group: $X = \begin{pmatrix}
0 & \alpha & \beta \\ 
0 & 0 & \gamma \\ 
0 & 0 & 0
\end{pmatrix}$ and $e^{tX} = \begin{pmatrix}
1 & t\alpha & t\beta + t^2 \alpha \gamma \frac{1}{2} \\ 
0 & 1 & t\gamma \\ 
0 & 0 & 1
\end{pmatrix}$  

Let $G$ be a matrix Lie group, and $X$ an element of its Lie
algebra. Then $e^X$ is an element of the identity component of $G$.

Let $G$ be a matrix Lie group with Lie algebra $\mathfrak{g}$. $X,Y \in \mathfrak{g}$ and $A \in G$.  
> $AXA^{-1} \in \mathfrak{g}$  
> $sX \in \mathfrak{g}$ for all real s  
> $X+Y \in \mathfrak{g}$  
> $XY - YX \in \mathfrak{g}$. (for physics convention $-i(XY - YX) \in \mathfrak{g}$)

**main theorem**
Let $G$ and $H$ be matrix Lie groups, with Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$, respectively. Suppose that $\phi : G \rightarrow H$ be a Lie group homomorphism. Then there exists a unique real linear map $\tilde{\phi} : \mathfrak{g} \rightarrow \mathfrak{h}$ such that $\phi(e^X) = e^{\tilde{\phi}(X)}$ for all $X \in \mathfrak{g}$ .
> $\tilde{\phi}(AXA^{-1}) = \phi(A) \tilde{\phi}(X) \phi(A)^{-1}$  
> $\tilde{\phi}([X,Y]) = [\tilde{\phi}(X), \tilde{\phi}(Y)]$  
> $\tilde{\phi}(X) = \left . \frac{d}{dx} \right \vert_{t=0} \phi(e^{tX})$

If $G, H, K$ are matrix Lie groups and $\phi : G \rightarrow H$, $\psi : G \rightarrow H$, then $\widetilde{\phi \circ \psi} = \tilde{\psi} \circ \tilde{\psi}$

Adjoint Mapping: the linear map $Ad A : \mathfrak{g} \rightarrow \mathfrak{g}$, defined as $Ad A(X) = AXA^{-1}$.  $Ad : G \rightarrow GL(\mathfrak{g})$ is group homomorphism.

Let $G$ be a matrix Lie group with Lie algebra $\mathfrak{g}$. Then there
exist a neighborhood $U$ of zero in $\mathfrak{g}$ and a neighborhood $V$ of $I$ in $G$ such that the
exponential mapping takes $U$ homeomorphically onto $V$ .

If $G$ is a connected matrix Lie group, then every element $A$ of $G$ can be written in the form $A = e^{X_1}e^{X_2} · · · e^{X_n}$ for some $X1,X2, · · ·Xn \in \mathfrak{g}$.

A finite-dimensional real or complex **Lie algebra** is a finite-dimensional real or complex vector space $\mathfrak{g}$, together with a map $[ ]$ from $\mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}$ , with the following properties:  
> $[]$ is bilinear.  
> $[X, Y ] = −[Y,X]$ for all $X, Y \in \mathfrak{g}$.  
> $[X, [Y,Z]] + [Y, [Z,X]] + [Z, [X, Y ]] = 0$ for all $X, Y,Z \in \mathfrak{g}$ (Jacobi identity).

The space $\mathfrak{g}l(n;R)$ of all $n \times n$ real matrices is a real Lie algebra.

Linear maps of $V$ into itself $\mathfrak{g}l(V)$ becomes a real or complex Lie algebra with the bracket operation $[A,B] = AB − BA$.

Every finite-dimensional real Lie algebra is isomorphic to a subalgebra of $\mathfrak{g}l(n;R)$. Every finite-dimensional complex Lie algebra is isomorphic to a (complex) subalgebra of \mathfrak{g}l(n;C)$. (This is
in contrast to the situation for Lie groups, where most but not all Lie groups are
matrix Lie groups.)

Let $\mathfrak{g}$ be a finite-dimensional real or complex Lie algebra, and let $X_1, · · · ,X_n$ be a basis for $\mathfrak{g}$ (as a vector space). Then for each $i, j$ $[X_i,X_j ]$ can be written uniquely in the form $[X_i,X_j ] = \sum_{k=1}^n c_{ijk}X_k$. $c_{ijk}$ are called **structure constants**.

$V$ is a finite-dimensional real vector space, then the **complexification** of $V$ , denoted $V_C$, is the space of formal linear combinations: $v_1 + iv_2$. With with $v_1, v_2 \in V$.  $V$ is the real subspace of $V_C$.

Let $\mathfrak{g}$ be a finite-dimensional real Lie algebra, and $\mathfrak{g}_C$ its complexification (as a real vector space). Then the bracket operation on $\mathfrak{g}$ has a unique extension to $\mathfrak{g}_C$ which makes $\mathfrak{g}_C$ into a complex Lie algebra. The complex Lie algebra $\mathfrak{g}_C$ is called the complexification of the real Lie algebra $\mathfrak{g}$. $$[X_1 + iX_2, Y_1 + iY_2] = ([X_1, Y_1] − [X_2, Y_2]) + i ([X_1, Y_2] + [X_2, Y_1])$$


$\mathfrak{gl}(n;\mathbb{R})_{\mathbb{C}} \simeq \mathfrak{gl}(n;\mathbb{C})$; 
$u(n)_{\mathbb{C}} \simeq \mathfrak{g}l(n;\mathbb{C})$; 
$\mathfrak{sl}(n;\mathbb{R})_{\mathbb{C}} \simeq \mathfrak{sl}(n;\mathbb{C})$; 
$so(n)_{\mathbb{C}} \simeq so(n;\mathbb{C})$

A given complex Lie algebra may have several non-isomorphic real forms.

**main theorem (inverse)**
1. Let $G$ and $H$ be matrix Lie groups, let $\phi_1,\phi_2 : G \rightarrow H$ be Lie group homomorphisms. And let $\tilde{\phi_1},\tilde{\phi_2} : \mathfrak{g} \rightarrow \mathfrak{h}$ be the associated Lie algebras homomorphisms. If $G$ is connected and $\tilde{\phi_1} = \tilde{\phi_2}$ then $\phi_1 = \phi_2$

2. Let $G$ and $H$ be matrix Lie groups with Lie algebras $\mathfrak{g},\mathfrak{h}$. Let $\tilde{\phi} : \mathfrak{g} \rightarrow \mathfrak{h}$ be a Lie algebra homomorphism. If $G$ is connected and simply connected, then there exists a unique Lie group homomorphism $\phi : G \rightarrow H$.  
> $\tilde{\phi}(AXA^{-1}) = \phi(A) \tilde{\phi}(X) \phi(A)^{-1}$  
> $\tilde{\phi}([X,Y]) = [\tilde{\phi}(X), \tilde{\phi}(Y)]$  
> $\tilde{\phi}(X) = \left . \frac{d}{dx} \right \vert_{t=0} \phi(e^{tX})$



Let $G$ be a connected matrix Lie group. A universal covering group of G (or just **universal cover**) is a connected, simply connected Lie group $\tilde{G}$, together with a Lie group homomorphism $\phi : \tilde{G} \to G$ (called the projection map) with the following properties:  
> $\phi$ maps $\tilde{G}$ onto $G$ (surjective)  
> There is a neighborhood $U$ of $I$ in  $\tilde{G}$ which maps homeomorphically under $\phi$ onto a neighborhood $V$ of $I$ in $G$.

If $G$ is any connected matrix Lie group, then a universal covering group $\tilde{G}$ of $G$ exists and is unique up to canonical isomorphism.

$G$ and $\tilde{G}$ have the same Lie algebra:  
Let $G$ be a connected matrix Lie group, $\tilde{G}$ its universal cover, and $\phi$ the projection map from $\tilde{G}$ to $G$. Suppose that $\tilde{G}$ is a matrix Lie group with Lie algebra $\tilde{ \mathfrak{g}}$. Then the associated Lie algebra map $\tilde{\phi} :\tilde{ \mathfrak{g}} \to \tilde{G}$ is an isomorphism.

Example: The universal cover of $S_1$ is $\mathbb{R}$, and the projection map is the map $x \to e^{ix}$.


finite-dimensional complex **representation** of $G$ is a Lie group homomorphism $\Pi : G \to GL(n;\mathbb{C})$

finite-dimensional complex representation of $\mathfrak{g}$ is a Lie algebra homomorphism $\pi$ of $\mathfrak{g} \to gl(n;\mathbb{C})$

$\pi(X) = \left . \frac{d}{dx} \right \vert_{t=0} \Pi (e^{tX})$

A representation with no non-trivial invariant subspaces is called **irreducible**.

A finite-dimensional representation of a group or Lie algebra, acting on a space $V$ , is said to be **completely reducible** if the following property is satisfied: Given an invariant subspace $W \subset V$ , and a second invariant subspace $U \subset W \subset V$ , there exists a third invariant subspace $\tilde{U} \subset W$ such that $U \cap \tilde{U} = \emptyset$ and $U \cup \tilde{U} = W$.

Let $\mathfrak{g}$ be a real Lie algebra, and $\mathfrak{g}_C$ its complexification. Then every finite-dimensional complex representation $\pi$ of $\mathfrak{g}$ has a unique extension to a (complex-linear) representation of $\mathfrak{g}_C$, also denoted $\pi$. The representation of $\mathfrak{g}_C$ satisfies: $\pi (X + iY ) = \pi (X) + i \pi (Y )$ for all $X \in \mathfrak{g}$.

Let $G$ be a matrix Lie group, let $H$ be a Hilbert space, and let $U(H)$ denote the group of unitary operators on $H$. Then a homomorphism $\Pi : G \to U(H)$ is called a **unitary representation** of $G$ if $\Pi$ satisfies the following
continuity condition: If $A_,A \in G$ and $A_n \to A$, then $\Pi(A_n)v \to \Pi(A)v$ for all $v \in H$. A unitary representation with no non-trivial closed invariant subspaces
is called irreducible.

A finite-dimensional completely reducible representation of a group or Lie algebra is equivalent to a direct sum of (one or more) irreducible representations.

Let $G$ be a matrix Lie group. Let $\Pi$ be a finite-dimensional unitary representation of $G$, acting on a finite-dimensional real or complex Hilbert space $V$ . Then $\Pi$ is completely reducible.

If $G$ is a finite group, then every finite-dimensional real or complex representation of $G$ is completely reducible.

If $G$ is a compact matrix Lie group, then every finite-dimensional real or complex representation of $G$ is completely reducible.

**Clebsch-Gordan** theory: Suppose $\Pi_1,\Pi_2$ are irreducible representations of a group $G$. If we regard
$\Pi_1 \otimes \Pi_2$ as a representation of $G$, it may no longer be irreducible. If it is not irreducible, one can attempt to decompose it as a direct sum of irreducible representations.

**Schur’s Lemma**: 
> Let $V$ and $W$ be irreducible real or complex representations of a group or Lie algebra, and let $\phi : V \to W$ be a
morphism. Then either $\phi = 0$ or $\phi$ is an isomorphism.  
> Let $V$ be an irreducible complex representation of a group or Lie algebra, and let $\phi : V \to V$ be a morphism of $V$ with itself. Then $\phi = \lambda I$, for some $\lambda \in \mathbb{C}$.  
> Let $V$ and $W$ be an irreducible complex representation of a group or Lie algebra, and let $\phi_1,\phi_2 : V \to W$ be non zero morphisms. Then $\phi_1 = \lambda \phi_2$, for some $\lambda \in \mathbb{C}$.

An irreducible complex representation of a commutative group or Lie algebra is one-dimensional.


## SU(2) and SO(3)

$SU(2)$ : $A = \begin{pmatrix}
\alpha & -\bar{\beta} \\ 
\beta & \bar{\alpha}
\end{pmatrix}$ where $\vert \alpha \vert ^2 + \vert \beta \vert ^2 = 1$ , that is we have $S^3$ in $\mathbb{R}^4$. compact, path connected and simply connected.

$SO(3)$ can be parametrized with $R(\phi,\vec{n})$, counterclockwise rotation about the axis through $\vec{n}$ by angle $\phi$. If $0 \le \phi \le \pi$, $\vec{n}$ and $\phi$ define exactly a unique rotation: except $R(\pi, \pm n)$ which are identical. We have a solid ball in $\mathbb{R}^3$ with radius $\pi$, Identity is the center of the ball, and antipodal points are glued together. compact, path connected but not simply connected.

[Dirac belt Trick](https://vimeo.com/62228139) (or hand rotation, twice)

Consider the space $V$ of all $2 \times 2$ complex matrices which are self-adjoint and have trace zero. And inner product as $\langle A,B \rangle = trace(AB)$. This is a three dimensional vector space. With orthonormal basis $\{ \sigma_x,\sigma_y,\sigma_z \}$. If $U \in SU(2)$ then $UAU^{-1} \in V$. $\Phi_U(A) \equiv UAU^{-1}$ is a linear map of $V$ into itself. More $\langle \Phi_U(A),\Phi_U(B) \rangle = \langle A,B \rangle$. $U \to \Phi_U$ is homomorphism and continuous.  
Thus $U \to \Phi_U$ is a Lie group homomorphism of $SU(2)$ into $SO(3)$ (in fact two to one map: $\Phi_U = \Phi_{-U}$).







## Clifford group


## Links

[An Elementary Introduction to Groups and Representations (Brian C. Hall)](https://arxiv.org/abs/math-ph/0005032)



## To Look

Symplectic group

Orbit-stabilizer theorem and Burnside's lemma

homomorphism theorems





```python
from sympy import *
init_printing(use_latex='mathjax')
from sympy.physics.quantum.dagger import Dagger
a, b, c, d = symbols('a b c d')
m = Matrix([[a,b],[c,d]])
display(m)
Dagger(m)*m, m*Dagger(m)


```


$$\left[\begin{matrix}a & b\\c & d\end{matrix}\right]$$





$$\left ( \left[\begin{matrix}a \overline{a} + c \overline{c} & b \overline{a} + d \overline{c}\\a \overline{b} + c \overline{d} & b \overline{b} + d \overline{d}\end{matrix}\right], \quad \left[\begin{matrix}a \overline{a} + b \overline{b} & a \overline{c} + b \overline{d}\\c \overline{a} + d \overline{b} & c \overline{c} + d \overline{d}\end{matrix}\right]\right )$$




```python
print('aa')
1+1

```

    aa





    2




```python

```

8ee2a24dc7553144d5916e8f5e252c412c21f7e2
ipynb
Jupyter Notebook
Math/IntroToGroupsLie.ipynb
gate42qc/seminars
35ff77b902d9c2ede619fd6e2d9c3e80d20d78de
MIT
2018-12-07T10:02:06.000Z
2019-11-24T19:30:03.000Z
Math/IntroToGroupsLie.ipynb
gate42qc/seminars
35ff77b902d9c2ede619fd6e2d9c3e80d20d78de
MIT
Math/IntroToGroupsLie.ipynb
gate42qc/seminars
35ff77b902d9c2ede619fd6e2d9c3e80d20d78de
MIT
2019-08-22T12:07:40.000Z
2019-08-22T12:07:40.000Z
Qwen/Qwen-72B
1. YES
2. YES