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Category Archives: Lie Groups and Lie Algebras
The Representation of $\mathrm{SU}(2)$
Let $\mathcal{H}_j$ be the space of polynomial functions on $\mathbb{C}^2$ that are homogemeous of degree $2j$. An element in $\mathcal{H}_j$ is a polynomial in complex variables $x$ and $y$ that is a linear combination of polynomials $x^py^q$ where $p+q=2j$. $\mathcal{H}_j$ … Continue reading
Lie Group Actions and Lie Group Representations
$\mathrm{SO}(3)$ acts on $\mathbb{R}^3$ meaning that each element of $\mathrm{SO}(3)$ defines a linear transformation (rotation) of $\mathbb{R}^3$. So we can say that $\mathrm{SO}(3)$ describes the rotational symmetry of $\mathbb{R}^3$. Definition. A group $G$ is said to act on a vector … Continue reading
More Examples of Lie Groups: 3Sphere as a Lie group, The 3Dimensional Heisenberg Group
3Sphere $S^3$ as a Lie Group Consider 4 elements $1,i,j,k$ satisfying the following relation \begin{align*} 1^2&=1,\ i^2=j^2=k^2=1,\\ 1i&=i1=i,\ 1j=j1=j,\ 1k=k1=k,\\ ij&=ji=k,\ jk=kj=i,\ ki=ik=j. \end{align*} Let $\mathbb{H}$ be the algebra spanned by $1,i,j,k$ over $\mathbb{R}$ $$\mathbb{H}=\{a1+bi+cj+dk:a,b,c,d\in\mathbb{R}\}.$$ Then $\mathbb{H}\cong\mathbb{R}^4$ as a vector … Continue reading
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Lie Brackets (for $n\times n$ Matrices)
Definition. Given any two $n\times n$ matrices $A$ and $B$ define $$[A,B]:=ABBA.$$ $[\ ,\ ]$ is a bilinear operator on the Lie algebra $\mathfrak{gl}(n)$ of the general linear group $\mathrm{GL}(n)$ and is called Lie bracket. Note that with $[\ ,\ … Continue reading
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The Lorentz Group
Let $\mathbb{L}^4$ be $\mathbb{R}^4$ with the Lorentzian inner product $\langle\ ,\ \rangle$ defined by $$\langle v,w\rangle=v^0w^0+v^1w^1+v^2w^2+v^3w^3$$ for $v={}^t(v^0,v^1,v^2,v^3),w={}^t(w^0,w^1,w^2,w^3)\in\mathbb{R}^4$. In particular, $$v^2=(v^0)^2+(v^1)^2+(v^2)^2+(v^3)^2.$$ $\mathbb{L}^4$ is called the Minkowski $4$spacetime or simply 4spacetime. In physics, $\mathbb{L}^4$ is commonly denoted by $\mathbb{R}^{3+1}$. Another common … Continue reading
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Orthogonal Group $\mathrm{O}(n)$ and Symmetry
Denote by $\langle\ ,\ \rangle$ the standard Euclidean inner product in $\mathbb{R}^n$. Then for any $v,w\in\mathbb{R}^n$, $$\langle v,w\rangle={}^tvw.$$ Definition. A bijective map $A: \mathbb{R}^n\longrightarrow\mathbb{R}^n$ is said to be an isometry if it preserves the inner product $\langle\ ,\ \rangle$ i.e. … Continue reading
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Quantum Angular Momentum in $\mathbb{R}^{2+2}$ and $\mathfrak{su}(1,1)$ Representation
It can be shown that quantum angular momentum \begin{align*} L_x&=i\hbar\left(y\frac{\partial}{\partial z}z\frac{\partial}{\partial y}\right)\\ L_y&=i\hbar\left(z\frac{\partial}{\partial x}x\frac{\partial}{\partial z}\right)\\ L_z&=i\hbar\left(x\frac{\partial}{\partial y}y\frac{\partial}{\partial x}\right) \end{align*} can be obtained purely mathematically by $\mathfrak{su}(2)$ Lie algebra representation as discussed here. Since $\mathfrak{su}(2)$ representation contains information on the symmetry … Continue reading
Quantum Angular Momentum and $\mathfrak{su}(2)$ Representation
In classical mechanics, the angular momentum of a body is given by $$L=r\times p$$ where $r$ and $p$ denote radius arm and linear momentum respectively. In quantum mechanics, the angular momentum of a spinning particle can be obtained by replacing … Continue reading
The Lie Algebra of the Orthogonal Group $\mathrm{O}(n)\ (\mathrm{SO}(n))$
It can be easily shown that $${\rm SO}(2)=\left\{\left(\begin{array}{cc} \cos\theta & \sin\theta\\ \sin\theta & \cos\theta \end{array} \right): \theta\in[0,2\pi)\right\}\cong{\rm S}^1=\{e^{i\theta}: \theta\in[0,2\pi)\}.$$Let $\gamma(t)=\left(\begin{array}{cc} \cos\theta(t) & \sin\theta(t)\\ \sin\theta(t) & \cos\theta(t) \end{array} \right)\in\mathrm{SO}(2)$ with $\theta(0)=0$ and $\dot\theta(0)\ne 0$. Then $\gamma(t)$ be a differentiable (regular) curve … Continue reading
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Matrix Lie Groups
Definition. A group $(G,\cdot,{}^{1},e)$ is a Lie group if $G$ is also a differentiable manifold and the binary operation $\cdot: G\times G\longrightarrow G$ and the unary operation (inverse) ${}^{1}: G\longrightarrow G$ are smooth maps. A subgroup of a Lie group … Continue reading
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