THE QUANTUM sl(n, ℂ) REPRESENTATION THEORY AND ITS APPLICATIONS

Abstract

Abstract. In this paper, we study the quantum sl(n) representation category using the web space. Specially, we extend sl(n) web space for n ≥ 4 as generalized Temperley-Lieb algebras. As an application of our study, we find that the HOMFLY polynomial Pn(q) specialized to a one variable polynomial can be computed by a linear expansion with respect to a presentation of the quantum representation category of sl(n). Moreover, we correct the false conjecture [30] given by Chbili, which addresses the relation between some link polynomials of a periodic link and its factor link such as Alexander polynomial (n = 0) and Jones polynomial (n = 2) and prove the corrected conjecture not only for HOMFLY polynomial but also for the colored HOMFLY polynomial specialized to a one variable polynomial.

On skein relations in class S theories

“…Our guiding principle is that they are described by the structure of the quantum groups underlying both the q-deformed Yang-Mills and the Liouville/Toda CFT, and that the skein relations are universal under an appropriate parameter identification. The skein relations exhibited below already appear in the mathematical works [17,31,32], up to the overall factors and the changes in conventions. Skein relations introduce equivalence relations among all possible networks, and it would be extremely useful if we can pick a natural representative element out of a given equivalence class of networks allowing linear combinations of networks.…”

$U_q(\mathfrak{sl}_n)$ web models and $\mathbb{Z}_n$ spin interfaces

“…so that we have effectively x : 0, n − 1 → C with x 0 = 1. The group Z n acts by cyclically permuting the values 0, n − 1 of all spins, and this action leaves (28) invariant as it only depends on the differences σ i − σ j .…”

“…They are relevant for the physical description of domain walls in spin systems [18][19][20], where the spin can take more than two values, or for certain network models motivated by topological phases [21][22][23]. In parallel, the mathematical literature has seen the definition of algebraic structures whose geometrical representation takes the form of bifurcating objects [24][25][26][27][28][29][30][31] also known as "spiders". To be precise, the algebra underlying the description of loops is the celebrated Temperley-Lieb algebra [32] and its close cousins (with higher-spin representations [34,35], and versions with dilution [7,36], with colours [37,38] or the fully-packed versions [39][40][41]).…”

“…The simplest example of an algebraic construction accounting for bifurcations is the so-called Kuperberg spider [25], see an example on the right of Figure 1. The study of such structures seems however to have been driven by rather formal motivations, such as classification of invariants of the quantum group U q (sl n ) with n ≥ 3 in its tensor-product representations [25,27,31], and for construction of isotopy invariants of links and knots generalising the Jones polynomials [26,28,30]. The usual loop case, described by the Temperley-Lieb algebra, is related to U q (sl 2 ) where the Temperley-Lieb diagrams classify the invariants in a certain tensor power of the fundamental representation of U q (sl 2 ).…”

“…In this paper, we follow the diagrammatical formalism of Cautis-Kamnitzer-Morrison for sl n calculus from [31]. In the literature there are a few other related, though not necessarily equivalent, formalisms [26][27][28], and we choose the one given in [31] mainly for practical reasons, to make physical properties (like CPT invariance) of our models evident. Moreover, the formalism of Cautis-Kamnitzer-Morrison becomes quite crucial when we study open variants of our models with boundary defects.…”