论文标题
从中间卡西米尔不变的谎言代数的多项式代数构建
Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras
论文作者
论文摘要
我们提出了一个系统的过程，以构建来自（半imple或non-semisimple）的中间casimir不变性的多项式代数，lie代数$ \ mathfrak {g} $。在这种方法中，我们在$ \ mathfrak {g} \ oplus \ mathfrak {g} \ oplus \ oplus \ mathfrak {g} $的包围代数中处理显式多项式。我们提出明确的例子，以说明这些谎言代数如何显示不同的行为，并可能导致Abelian代数，二次代数或更复杂的结构，涉及高阶嵌套换向器。在此框架内，我们还展示了Levi Docomposable Lie代数的Levi因子的虚拟副本如何用作构造多项式代数的“副本”的工具。在文献中提出了与代数哈密顿量相关的多项式代数的不同方案，其中包括使用各种类型的换向物。目前的方法是不同的，并且依赖于在包裹的代数$ \ Mathcal {u}（\ Mathfrak {g} \ oplus \ Mathfrak \ Mathfrak {G} \ oplus \ Mathfrak \ Mathfrak {g}）$的构造中。
We propose a systematic procedure to construct polynomial algebras from intermediate Casimir invariants arising from (semisimple or non-semisimple) Lie algebras $\mathfrak{g}$. In this approach, we deal with explicit polynomials in the enveloping algebra of $\mathfrak{g} \oplus \mathfrak{g} \oplus \mathfrak{g}$. We present explicit examples to show how these Lie algebras can display different behaviours and can lead to Abelian algebras, quadratic algebras or more complex structures involving higher order nested commutators. Within this framework, we also demonstrate how virtual copies of the Levi factor of a Levi decomposable Lie algebra can be used as a tool to construct "copies" of polynomial algebras. Different schemes to obtain polynomial algebras associated to algebraic Hamiltonians have been proposed in the literature, among them the use of commutants of various type. The present approach is different and relies on the construction of intermediate Casimir invariants in the enveloping algebra $\mathcal{U}(\mathfrak{g} \oplus \mathfrak{g} \oplus \mathfrak{g})$.
