Comments about quantum symmetries of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.gif" display="inline" overflow="scroll"><mml:mstyle mathvariant="italic"><mml:mi>SU</mml:mi></mml:mstyle><mml:mrow><mml:mo>(</mml:mo><mml:mn>3</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> graphs

Coquereaux, Robert; Hammaoui, D.; Schieber, Gil; Tahri, E. H.

doi:10.1016/j.geomphys.2006.03.002

Cited by 15 publications

(35 citation statements)

References 42 publications

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“…of Fig. 22: However, as we know, σ ⊗ σ = σ (1,0) ⊗ σ (0,1) = σ (0,0) ⊕ σ (1,1) , in terms of highest weights labels (or, in terms of quantum dimensions: [3]×[3] = [1]+[2] [4], or, classically, 3×3 = 1+8). So, from σ ⊗ σ we have two intertwiners, respectively to C and to σ (1,1) .…”

Section: The Elementary Versionmentioning

confidence: 99%

Notes on TQFT Wire Models and Coherence Equations for SU(3) Triangular Cells

Coquereaux¹

2010

SIGMA

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Abstract. After a summary of the TQFT wire model formalism we bridge the gap from Kuperberg equations for SU (3) spiders to Ocneanu coherence equations for systems of triangular cells on fusion graphs that describe modules associated with the fusion category of SU (3) at level k. We show how to solve these equations in a number of examples.Key words: quantum symmetries; module-categories; conformal field theories; 6j symbols 2010 Mathematics Subject Classification: 81R50; 81R10; 20C08; 18D10 ForewordStarting with the collection of irreducible integrable representations (irreps) of SU (3) at some level k (constructed in the framework of affine algebras or in the framework of quantum groups at roots of unity), the problem is to decide whether a graph encoding the action of these irreps actually defines a "healthy fusion graph" associated with a bona fide SU (3) nimrep, i.e. a module over a particular kind of fusion category. This is done by associating a complex number (a "triangular cell") with every elementary triangle of the given graph in such a way that their collection (a "self-connection") obeys a system of non trivial quadratic and quartic equations, called Ocneanu coherence equations, that can be themselves derived from another set of equations (sometimes called Kuperberg equations) describing relations between the intertwiners of the underlying fusion category. One issue is to describe and derive the coherence equations themselves. Another issue is to use them on the family of examples giving rise to SU (3) nimreps. Both problems are studied in this article.Our paper consists of two largely independent parts. The first is a set of notes dealing with TQFT graphical models (wire models, spiders, etc.). Our motivation for this part was to show how the Ocneanu coherence equations for triangular cells of fusions graphs could be deduced from the so-called Kuperberg relations for SU (3), something that does not seem to be explained in the literature. This section was then enlarged in order to set the discussion in the larger framework of graphical TQFT models and to discuss some not so well known features that show up when comparing the SU (2) and SU (3) situations. Despite its size, this first part should not be considered as a general presentation of the subject starting from first principles, this would require a book, not an article. The second part of the paper deals about the coherence equations themselves. In particular we show, on a selection of examples chosen among quantum subgroups (module-categories) of type SU (3), how to solve them in order to obtain a self-connection on the corresponding fusion graphs.

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Section: The Elementary Versionmentioning

confidence: 99%

Notes on TQFT Wire Models and Coherence Equations for SU(3) Triangular Cells

Coquereaux¹

2010

SIGMA

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“…The period (read vertically) is 2 × 5 since N = g + k = 2 + 3 = 5. Horizontally one recognizes the Dynkin diagram A 4 of SU (5). The chosen root is located where the "2" stands.…”

Section: On the Su(3) Classification (Reminders)mentioning

confidence: 99%

Theta Functions for Lattices of SU(3) Hyper-Roots

Coquereaux

2018

Experimental Mathematics

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We recall the definition of the hyper-roots that can be associated to modules-categories over fusion categories defined by the choice of a simple Lie group G together with a positive integer k. This definition was proposed in 2000, using another language, by Adrian Ocneanu. If G = SU(2), the obtained hyper-roots coincide with the usual roots for ADE Dynkin diagrams. We consider the associated lattices when G = SU(3) and determine their theta functions in a number of cases; these functions can be expressed as modular forms twisted by appropriate Dirichlet characters. arXiv:1708.00560v1 [math.QA] 2 Aug 2017The ADE correspondence between indecomposable module-categories of type SU(2) and simplylaced Dynkin diagrams was first obtained by theoretical physicists in the framework of conformal field theories (classification of modular invariant partition functions for the WZW models of type SU(2), [1], [8]). Its relation with subfactors was studied in [19] and it was set in a categorical framework by [16,24]. In plain terms, the diagrams encoding the action of the fundamental representation of SU(2) at level k (which is classically 2-dimensional) on the simple objects of the various module-categories existing at that level, are the Dynkin diagrams describing the simplylaced simple Lie groups with Coxeter number k + 2.At a deeper level, there is a correspondence between fusion coefficients of the SU(2) modulecategory described by a Dynkin diagram E and the inner products between all the roots of the simply-laced Lie group associated with the same Dynkin diagram. In the non-ADE cases one can also define the action of an appropriate ring on modules associated with the chosen Dynkin diagrams and still obtain a correspondence between structure coefficients describing this action and inner products between roots -one has only to introduce scaling coefficients in appropriate places.The correspondence relating fusion coefficients for module-categories of type SU(2) and inner products between weights and/or roots of root systems was clearly stated (but not much discussed) in [20]; in a different context, some of its aspects were already present in the article [10]. The correspondence was used and described in some detail in one section of [3]. As observed in [20], one can start from the SU(2) fusion categories and their modules 1 to recover or define the usual root systems, and associate with each of them a periodic quiver describing, in particular, the inner products between all the roots. It is also observed in the same reference that the construction can be generalized: replacing SU(2) by an arbitrary simple Lie group G leads, for every choice of a module-category E of type 2 G, to a system of "higher roots" that we call "hyper-roots of type G". Usual root systems are therefore hyper-root systems of type SU(2). A usual root system gives rise, in particular, to an Euclidean lattice. The same is true for hyper-root systems. Given a lattice, one may consider its theta series whose n th coefficient gives the number of vectors of g...

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“…Further compatibility conditions have also to be satisfied and can be used to check the results, or to determine the remaining coefficients (for degenerate cases). One of these conditions read [23,9]:…”

Section: Determination Of the Ocneanu Algebra O Xmentioning

confidence: 99%

From modular invariants to graphs: the modular splitting method

Isasi¹,

Schieber²

2007

J. Phys. A: Math. Theor.

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We start with a given modular invariant M of a two dimensional su(n) k conformal field theory (CFT) and present a general method for solving the Ocneanu modular splitting equation and then determine, in a step-by-step explicit construction, 1) the generalized partition functions corresponding to the introduction of boundary conditions and defect lines; 2) the quantum symmetries of the higher ADE graph G associated to the initial modular invariant M. Notice that one does not suppose here that the graph G is already known, since it appears as a by-product of the calculations. We analyze several su(3) k exceptional cases at levels 5 and 9.

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Comments about quantum symmetries of SU(3) graphs

Cited by 15 publications

References 42 publications

Notes on TQFT Wire Models and Coherence Equations for SU(3) Triangular Cells

Notes on TQFT Wire Models and Coherence Equations for SU(3) Triangular Cells

Theta Functions for Lattices of SU(3) Hyper-Roots

From modular invariants to graphs: the modular splitting method

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