Consider $\mathbb{R}^2$ as an Abelian Lie algebra and let $c$ be a non-zero antisymmetric bilinear form on $\mathbb{R}^2$. We then define the three-dimensional Heisenberg Lie algebra $\mathbb{R}^3=\mathbb{R}^2 \oplus \mathbb{R}$ whose elements are pairs $(X,\lambda)$ endowed with Lie bracket $[(X,\lambda),(U,\mu)]\equiv ([X,Y],c(X,Y))$. I have a question: Why do study Heisenberg Lie algebra or Heisenberg Lie group (which is the Lie group corresponding the the Heisenberg Lie algebra)? Are they universal covering or universal central extension of the original Lie algebra $\mathbb{R}^2$ (or original group)?
Asked
Active
Viewed 61 times
