论文标题
陈述的SL（N）-Skein模块和代数
Stated SL(n)-Skein Modules and Algebras
论文作者
论文摘要
我们开发了一种陈述的SL（n）-skein模块，$ s_n（m，n）的理论，$ 3 manifolds $ m $标记为间隔$ n $的边界。它们由$ n $ webs的线性组合组成，其中$ n $的结尾是受雷希蒂基·图拉耶夫（Reshetikhin-Turaev）理论关系启发的skein关系。 我们证明，将$ m $沿磁盘切割，从而产生$ 3 $ -Manifold $ m'$产生同构$ s_n（m）\ to s_n（m'）$。该结果允许通过其作品的绞线模块分析$ 3 $ manifolds的绞线模块。 陈述的绞线模块的理论特别丰富，对于厚厚的表面$ m =σ\ times（-1,1），$，在其情况下，$ s_n（m）$是一个代数，用$ s_n（σ）表示。$我们证明，我们证明了理想的bigon的skein代数是$ o_q（n）$ o_q（n）$ geefters $ o_q（n）$ gee and geefter and geefter and geefter and geefters $ $ o_q（sl（n））上的counit，antipode和共同结构。 此外，我们表明，在标记附近对增稠的Bigon的分裂定义了$ o_q（sl（n））$ - comodule结构$ s_n（m），$或双重，$ u_q（sl_n）$ - 模块结构。此外，我们表明，沿三角形的两个侧面粘合的表面的绞线代数$σ_1，σ_2$是同构的，与编织的张量产品$ s_n（σ_1）\下划线{\ otimes} s_n（Σ_2）这些结果允许对量子群理论（例如编织产品和Majid的trans变操作）的进一步概念进行几何解释。 我们证明，在$ rep \，u_q（sl_n）$中具有系数的表面的分解同源性等同于$ s_n（σ）$的左模块的类别。我们还讨论了与Alekseev-Schomerus的量子模量空间的关系。 最后，我们表明，对于带边界的表面$σ$，$ s_n（σ）$是一个自由模块，其基础是从kashiwara-lusztig canonical bases引起的。
We develop a theory of stated SL(n)-skein modules, $S_n(M,N),$ of 3-manifolds $M$ marked with intervals $N$ in their boundaries. They consist of linear combinations of $n$-webs with ends in $N$, considered up to skein relations inspired by the relations of the Reshetikhin-Turaev theory. We prove that cutting $M$ along a disk resulting in a $3$-manifold $M'$ yields a homomorphism $S_n(M)\to S_n(M')$. That result allows to analyze the skein modules of $3$-manifolds through the skein modules of their pieces. The theory of stated skein modules is particularly rich for thickened surfaces $M=Σ\times (-1,1),$ in whose case, $S_n(M)$ is an algebra, denoted by $S_n(Σ).$ We prove that the skein algebra of the ideal bigon is $O_q(SL(n))$ and that it provides simple geometric interpretations of the product, coproduct, counit, the antipode, and the cobraided structure on $O_q(SL(n)).$ Additionally, we show that a splitting of a thickened bigon near a marking defines a $O_q(SL(n))$-comodule structure on $S_n(M),$ or dually, an $U_q(sl_n)$-module structure. Furthermore, we show that the skein algebra of surfaces $Σ_1, Σ_2$ glued along two sides of a triangle is isomorphic with the braided tensor product $S_n(Σ_1)\underline{\otimes} S_n(Σ_2)$ of Majid. These results allow for a geometric interpretation of further concepts in the theory of quantum groups, for example, of the braided products and of Majid's transmutation operation. We prove that the factorization homology of surfaces with coefficients in $Rep\, U_q(sl_n)$ is equivalent to the category of left modules over $S_n(Σ)$. We also discuss the relation with the quantum moduli spaces of Alekseev-Schomerus. Finally, we show that for surfaces $Σ$ with boundary, $S_n(Σ)$ is a free module with a basis induced from the Kashiwara-Lusztig canonical bases.
