论文标题
$ \ mathfrak {sl} _2 $
Higher-spin quantum and classical Schur-Weyl duality for $\mathfrak{sl}_2$
论文作者
论文摘要
众所周知，$ u_q（\ Mathfrak {sl} _2）$ - 对其基本模块的$ n $ fold张量产物的换向代数对temperley-lieb代数tl $ _n（n n（n）与fugacity参数$ camem $ cament camem of temperley-lieb algebra tl $ _n（即，当$ q $不是统一的根源时，或$ n $足够小）。此外，简单的$ u_q（\ mathfrak {sl} _2）$ - 在$ n $ fold-fold Tensor产品模块的直接和themum分解中出现的模块是与Temperley-Lieb代数的模块一对一的对应关系。这种双重交换性属性称为量子schur-weyl二元性。 在本文中，我们详细研究了这种双重性。我们证明，$ u_q（\ mathfrak {sl} _2）$ - 对任何通用类型的One张量产品模块模块的操作都是对代数的同构，我们称之为valencred templey templeybe templeybemperley-lieb-lieb代数代数tl $_ς（ν）$。这对应于具有较高自旋的表示形式，这导致了templeley-lieb图中的价值（或颜色）。我们建立了表现出这种二元性的详细直接分解，并找到适合具体计算的明确基础，这在应用中很重要。我们还包括一个双重交换类型属性，用于不同$ u_q（\ mathfrak {sl} _2）$ - 模块之间的同构属性，该模块由VANEDER图实现。该图计算让人联想到Kauffman的重耦理论，以及Penrose和Frenkel \＆Khovanov开发的图形方法。结果还包含标准的量子schur-weyl二重性作为特殊情况，当专门为$ q \ rightarrow 1 $时，这意味着lie代数$ \ mathfrak {sl} _2 _2（\ mathbb {c}）$ and frobenius-schur-weyl二元性。
It is well-known that the commutant algebra of the $U_q(\mathfrak{sl}_2)$-action on the $n$-fold tensor product of its fundamental module is isomorphic to the Temperley-Lieb algebra TL$_n(ν)$ with fugacity parameter $ν= -q - q^{-1}$ (at least in the generic case, i.e., when $q$ is not a root of unity, or $n$ is small enough). Furthermore, the simple $U_q(\mathfrak{sl}_2)$-modules appearing in the direct-sum decomposition of the $n$-fold tensor product module are in one-to-one correspondence with those of the Temperley-Lieb algebra. This double-commutant property is referred to as quantum Schur-Weyl duality. In this article, we investigate such a duality in great detail. We prove that the commutant of the $U_q(\mathfrak{sl}_2)$-action on any generic type-one tensor product module is isomorphic to a diagram algebra that we call the valenced Temperley-Lieb algebra TL$_ς(ν)$. This corresponds to representations with higher spin, which results in the need of valences (or colors) in the Temperley-Lieb diagrams. We establish detailed direct-sum decompositions exhibiting this duality and find explicit bases amenable to concrete calculations, important in applications. We also include a double-commutant type property for homomorphisms between different $U_q(\mathfrak{sl}_2)$-modules, realized by valenced diagrams. The diagram calculus is reminiscent to Kauffman's recoupling theory and the graphical methods developed among others by Penrose and Frenkel \& Khovanov. The results also contain the standard quantum Schur-Weyl duality as a special case, and when specialized to $q \rightarrow 1$, imply the classical Frobenius-Schur-Weyl duality for the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$ and a higher-spin version thereof.
