论文标题
小量子组的半量子标志品种和表示形式
The half-quantum flag variety and representations for small quantum groups
论文作者
论文摘要
考虑一个几乎简单的代数G组，并选择统一Q的复杂根。我们研究了半量子标志品种上的准搭扣杆$ \ mathscr {x} _q $的类别，在经典标志品种g/b上，该类别本身形成了张量类别的包装。我们证明，Q处的小量子组表示的类别完全忠实地嵌入了$ \ mathscr {x} _q $的全局，而$ \ mathscr {x} _q $ of G/b的$ \ mathscr {x} _q $ over g/b的纤维恢复了小型量子borelels的张量。这些关系既处于亚洲和衍生水平。随后，从小量子组到其鲍尔尔的缩小参数在$ \ mathscr {x} _q $上以“光纤检查”参数的形式出现为“纤维检查”参数。我们猜想$ \ Mathscr {x} _q $还包含弹簧分辨率的DG滑轮的类别，作为派生级别的完整单体子类别，因此在Springer分辨率和小量子组之间提供了单相对应关系。我们将这种猜想与弹簧分辨率的DG滑轮与量子组表示派生类别中的主要块之间的已知等效性联系起来[ABG04，BL07]。
Consider an almost-simple algebraic group G and a choice of complex root of unity q. We study the category of quasi-coherent sheaves $\mathscr{X}_q$ on the half-quantum flag variety, which itself forms a sheaf of tensor categories over the classical flag variety G/B. We prove that the category of small quantum group representations for G at q embeds fully faithfully into the global sections of $\mathscr{X}_q$, and that the fibers of $\mathscr{X}_q$ over G/B recover the tensor categories of representations for the small quantum Borels. These relationships hold both at an abelian and derived level. Subsequently, reduction arguments, from the small quantum group to its Borels, appear algebrogeometrically as "fiber checking" arguments over $\mathscr{X}_q$. We conjecture that $\mathscr{X}_q$ also contains the category of dg sheaves over the Springer resolution as a full monoidal subcategory, at the derived level, and hence provides a monoidal correspondence between the Springer resolution and the small quantum group. We relate this conjecture to a known equivalence between dg sheaves on the Springer resolution and the principal block in the derived category of quantum group representations [ABG04, BL07].
