2020/08/13 by Steven M. Flores, Flores, Steven M., Eveliina Peltola +1
Mathematics · Chemistry · #Algebraic structures and combinatorial models #Advanced Algebra and Geometry #Advanced NMR Techniques and Applications
paper · pdf · doi:10.48550/arxiv.2008.06038
It is well-known that the commutant algebra of the\nUq( mathfraksl2)-action on the n-fold tensor product of its fundamental\nmodule is isomorphic to the Temperley-Lieb algebra TLn(\ν) with fugacity\nparameter \ν = -q - q-1 (at least in the generic case, i.e., when q is\nnot a root of unity, or n is small enough). Furthermore, the simple\nUq( mathfraksl2)-modules appearing in the direct-sum decomposition of the\nn-fold tensor product module are in one-to-one correspondence with those of\nthe Temperley-Lieb algebra. This double-commutant property is referred to as\nquantum Schur-Weyl duality.\n In this article, we investigate such a duality in great detail. We prove that\nthe commutant of the Uq( mathfraksl2)-action on any generic type-one\ntensor product module is isomorphic to a diagram algebra that we call the\nvalenced Temperley-Lieb algebra TL_ varsigma(\ν). This corresponds to\nrepresentations with higher spin, which results in the need of valences (or\ncolors) in the Temperley-Lieb diagrams. We establish detailed direct-sum\ndecompositions exhibiting this duality and find explicit bases amenable to\nconcrete calculations, important in applications. We also include a\ndouble-commutant type property for homomorphisms between different\nUq( mathfraksl2)-modules, realized by valenced diagrams. The diagram\ncalculus is reminiscent to Kauffman's recoupling theory and the graphical\nmethods developed among others by Penrose and Frenkel & Khovanov. The results\nalso contain the standard quantum Schur-Weyl duality as a special case, and\nwhen specialized to q \→ 1, imply the classical Frobenius-Schur-Weyl\nduality for the Lie algebra mathfraksl2(\ℂ) and a higher-spin\nversion thereof.\n