# Monthly Archives: September 2011

## Lie Group and Lie Algebra Representations

Given a matrix Lie group $G$, a representation $\Pi$ of $G$ is a Lie group homomorphism $\Pi: G\longrightarrow\mathrm{GL}(V)$, where $V$ is a finite dimensional vector space and the general linear group $\mathrm{GL}(V)$ is the set of all linear isomorphisms of … 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

Posted in Lie Groups and Lie Algebras | 1 Comment

## 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

Posted in Lie Groups and Lie Algebras | 2 Comments

## Electrostatic Potential in a Hollow Cylinder

An electrostatic field $E$ (i.e. an electric field produced only by a static charge) is a conservative field, i.e. there exists a scalar potential $\psi$ such that $E=-\nabla\psi$. This is clear from Maxwell’s equations. Since there is no change of … Continue reading