标题： 陈述的绳结理论和双仿射赫克代数表示 显示英文标题

标题： Stated Skein Theory and Double Affine Hecke Algebra Representations

摘要： 在本论文中，我们通过陈述的辫群理论视角来探索双仿射海克代数（DAHAs）的表示理论。 在过去十年中，已有几项研究建立了辫群代数与DAHAs之间的牢固联系。 特别是，Samuelson证明了类型$A_1$ DAHA 的球形子代数可以作为环面边界上的Kauffman括号辫群代数的商来实现，$K_q(T^2 \setminus D^2)$。 由于$A_1$双仿射海克代数与它的球形子代数是Morita等价的，因此发现$K_q(T^2 \setminus D^2)$的模立即为我们提供了$A_1$ DAHA 的模。 陈述的辫群理论通过引入流形的边界分量来增强传统的Kauffman括号辫群理论，从而提供了诸如切除等额外性质，丰富了代数结构。 此外， Kauffman括号辫群代数嵌入到它们的陈述对应物中，表明陈述的辫群代数是Kauffman括号辫群代数的扩展。 我们利用这个扩展框架进一步发展$A_1$ DAHA 的表示理论。 在确定了所述的$T^2 \setminus D^2$的 skein 代数的生成元之后，我们将该代数嵌入到一个量子$6$-torus 中，并利用量子 tori 的良好表示论性质来构造一个 Laurent 多项式模块。 此外，由于$T^2$是任何纽结补的边界，我们讨论了如何从各种纽结构造一个更拓扑定义的模块，并为平凡纽结提供了一个显式例子。 这种方法建立在 Berest 和 Samuelson 的想法之上，他们证明了在纽结补的 Kauffman 折叠 skein 模块上存在一个自然的 DAHA 作用。 显示英文摘要

摘要： In this thesis, we explore the representation theory of double affine Hecke algebras (DAHAs) through the lens of stated skein theory. Over the past decade, there have been several works establishing robust connections between skein algebras and DAHAs. Particularly, Samuelson proved that a spherical subalgebra of the type $A_1$ DAHA can be realized as a quotient of the Kauffman bracket skein algebra of the torus with boundary, $K_q(T^2 \setminus D^2)$. Since the $A_1$ double affine Hecke algebra is Morita equivalent to its spherical subalgebra, discovering modules for $K_q(T^2 \setminus D^2)$ immediately provides us with modules for the $A_1$ DAHA. Stated skein theory enhances traditional Kauffman bracket skein theory by incorporating the boundary components of manifolds, thereby offering additional properties such as excision that enrich the algebraic structure. Furthermore, Kauffman bracket skein algebras embed into their stated counterparts, showing that stated skein algebras are extensions of Kauffman bracket skein algebras. We use this extended framework to further develop the representation theory of the $A_1$ DAHA. After identifying generators for the stated skein algebra of $T^2 \setminus D^2$, we embed this algebra into a quantum $6$-torus and leverage the nice representation-theoretic properties of quantum tori to construct a module of Laurent polynomials. Additionally, as $T^2$ is the boundary of any knot complement, we discuss how to construct a more topologically-defined module from various knots and provide an explicit example for the unknot. This approach builds upon the ideas of Berest and Samuelson, who showed that there exists a natural DAHA action on the Kauffman bracket skein module of knot complements.