非交换辛约化的交换实例
Abelian instances of nonabelian symplectic reduction
AI总结：
研究在哈密顿群作用下，非交换辛约化与交换子群辛约化等价的充要条件，并给出包括海森堡群在内的幂零李群族实例。
摘要：
设 $\mathbb{G}$ 是一个具有正规交换子群 $\mathbb{A}$ 的李群，$(M,\omega)$ 是一个具有哈密顿 $\mathbb{G}$ 作用的辛流形。我们研究 $\mathbb{G}$ 的辛约化与交换子群 $\mathbb{A}$ 的辛约化一致的条件。利用逐步约化框架（Marsden 等，Springer Notes in Math., 1913, (2007)），我们证明，在一个温和假设下，相应的约化空间是辛同构的当且仅当它们具有相同的维数。该假设和维数条件仅依赖于群 $\mathbb{G}$ 和 $\mathbb{A}$，以及执行 $\mathbb{G}$ 辛约化时的动量值 $\mu\in \mathfrak{g}^*$；特别地，它们与辛流形 $(M,\omega)$ 无关。然后，我们通过识别一大类幂零李群（包括经典 Carnot 群，如海森堡群和射流空间 $\mathcal{J}^k(\mathbb{R}^n,\mathbb{R}^m)$）来提供广泛的例子，对于这些群，两个约化空间在一般动量值下是辛同构的。
英文摘要：
Let $\mathbb{G}$ be a Lie group with a normal abelian subgroup $\mathbb{A}$, and let $(M,ω)$ be a symplectic manifold endowed with a Hamiltonian $\mathbb{G}$-action. We investigate conditions under which symplectic reduction by $\mathbb{G}$ coincides with the symplectic reduction by the abelian subgroup $\mathbb{A}$. Using the reduction-by-stages framework (Marsden et al Springer Notes in Math., 1913, (2007)), we prove that, under a mild assumption, the corresponding reduced spaces are symplectomorphic if and only if they have the same dimension. Both this assumption and the dimension condition depend only on the groups $\mathbb{G}$ and $\mathbb{A}$, and on the momentum value $μ\in \mathfrak{g}^*$ at which the symplectic reduction by $\mathbb{G}$ is performed; in particular, they are independent of the symplectic manifold $(M,ω)$. We then provide a broad class of examples by identifying a large family of nilpotent Lie groups, including classical Carnot groups such as the Heisenberg group and jet-space $\mathcal{J}^k(\mathbb{R}^n,\mathbb{R}^m)$, for which the two reduced spaces are symplectomorphic for generic momentum values.
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