论文标题
零卡西米尔的积极表示
Positive Representations with Zero Casimirs
论文作者
论文摘要
在本文中，我们基于Casimir操作员在某些Hilbert空间上的零元素构建了一个新的概括分裂量子群的概括。它是由在$ \ Mathcal {u} _Q（\ Mathfrak {sl}（2，2，\ Mathbb {r}）$兼容的最简单情况下，在最简单的情况下修改表示形式而产生的新观察。对于较高的等级，用$ \ Mathcal {u} _q（\ Mathfrak {g}）$的Chevalley Generator的极化获得了表示形式，这是在新的意识到的，它是普遍的lourent laurent多项式的特定偏度符号量的量子群集群。我们还明确计算了最大$ a_ {n-1} $变性表示的Casimir动作。$ \ Mathcal {U} _Q（\ Mathfrak {\ Mathfrak {g} _ \ Mathbb {r}）$用于基于中央参数的络合物的常规lie类型。
In this paper, we construct a new family of generalization of the positive representations of split-real quantum groups based on the degeneration of the Casimir operators acting as zero on some Hilbert spaces. It is motivated by a new observation arising from modifying the representation in the simplest case of $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$ compatible with Faddeev's modular double, while having a surprising tensor product decomposition. For higher rank, the representations are obtained by the polarization of Chevalley generators of $\mathcal{U}_q(\mathfrak{g})$ in a new realization as universally Laurent polynomials of a certain skew-symmetrizable quantum cluster algebra. We also calculate explicitly the Casimir actions of the maximal $A_{n-1}$ degenerate representations of $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$ for general Lie types based on the complexification of the central parameters.
