# 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: 3-Sphere as a Lie group, The 3-Dimensional Heisenberg Group

3-Sphere $S^3$ 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