论文标题
海森堡 - 韦尔代数的谎言结构
Lie structure of the Heisenberg-Weyl algebra
论文作者
论文摘要
作为协会代数，Heisenberg-Weyl代数$ \ Mathcal {H} $由两个元素$ a $，$ b $生成，约束$ ab-ba = 1 $。但是，作为谎言代数，通常的换向器用作谎言括号，元素$ a $和$ b $无法生成整个空间$ \ mathcal {h} $。我们确定了$ \ Mathcal {h} $的非努力但可解决的谎言sibalgebra $ \ mathfrak {g} $，为此，使用了自由代数的基础理论中的一些事实，我们为生成器和关系提供了介绍。在本演讲中，我们表明，对于某些代数同构$φ：\ MATHCAL {H} \ LONGRIGHTROW \ MATHCAL \ MATHCAL {H} $，lie代数$ \ Mathcal {H h} $是由$ \ Mathfrak {g} $的生成$ $ $ $ $ n of y MATH $ n of fer $ \ mathfrak {g} $，$φ（\ mathfrak {g}）$和$ \ left [\ mathfrak {g}，φ（\ mathfrak {g}）\ right] $。
As an associative algebra, the Heisenberg-Weyl algebra $\mathcal{H}$ is generated by two elements $A$, $B$ subject to the relation $AB-BA=1$. As a Lie algebra, however, where the usual commutator serves as Lie bracket, the elements $A$ and $B$ are not able to generate the whole space $\mathcal{H}$. We identify a non-nilpotent but solvable Lie subalgebra $\mathfrak{g}$ of $\mathcal{H}$, for which, using some facts from the theory of bases for free Lie algebras, we give a presentation by generators and relations. Under this presentation, we show that, for some algebra isomorphism $φ:\mathcal{H}\longrightarrow\mathcal{H}$, the Lie algebra $\mathcal{H}$ is generated by the generators of $\mathfrak{g}$, together with their images under $φ$, and that $\mathcal{H}$ is the sum of $\mathfrak{g}$, $φ(\mathfrak{g})$ and $\left[ \mathfrak{g},φ(\mathfrak{g})\right]$.
