## Modular data: the algebraic combinatorics of conformal field theory (2001)

Citations: | 36 - 5 self |

### BibTeX

@MISC{Gannon01modulardata:,

author = {Terry Gannon},

title = {Modular data: the algebraic combinatorics of conformal field theory },

year = {2001}

}

### OpenURL

### Abstract

### Citations

517 | Conformal Field Theory - Francesco, Mathieu, et al. - 1997 |

405 |
Quantum groups
- Kassel
- 1995
(Show Context)
Citation Context ...ays involves combining a given (inadequate) algebraic structure with its dual in some way. A general categorical interpretation of quantum double is the centre construction, described for instance in =-=[64]-=-; it assigns to a tensor category a braided tensor category. It would be interesting to interpret this construction at the more base level of fusion ring — e.g. as a general way for obtaining self-dua... |

356 |
Infinite dimensional Lie algebras, 3rd Ed
- Kac
- 1990
(Show Context)
Citation Context ...1 index-3 subgroup of SL2(Z). Nothing essential though is lost, so the definition of modular data should be broadened to include all integral lattice examples. Example 2: Kac-Moody algebras. See e.g. =-=[60,63]-=- for the basics of Kac-Moody algebras. The source of some of the most interesting modular data are the affine nontwisted KacMoody algebras X (1) r . The simplest way to construct affine algebras is to... |

314 |
Boundary Conditions, Fusion Rules And The Verlinde Formula
- Cardy
- 1989
(Show Context)
Citation Context ...(short for ‘nonnegative integer representation’ [11]) or equivalently fusion graph. These originally arose in two a 25priori unrelated contexts: the analysis, starting with Cardy’s fundamental paper =-=[15]-=-, of boundary RCFT; and Di Francesco–Zuber’s largely empirical attempt [23] to understand and generalise the A-D-E metapattern appearing in A (1) modular invariants, by attaching graphs to each confor... |

279 |
Vertex Operator Algebras and the Monster
- Frenkel, Lepowsky, et al.
- 1988
(Show Context)
Citation Context .... Extensions of this picture to representations of higher genus mapping class groups is discussed in [69] for CFT and [2,81] for TFT, but there is much more work to do here. Example 5: VOAs. See e.g. =-=[36,60]-=- for the basic facts about VOAs; the review article [44] explains how VOAs naturally arise in CFT. Another very general source of modular data comes from vertex operator algebras (VOAs), a rich algebr... |

245 |
Character Theory of Finite Groups
- Isaacs
- 1994
(Show Context)
Citation Context ...r rather how modular data should be generalised to accommodate them, is not completely understood at this time. Example 3: Finite groups. The relevant aspects of finite group theory are given in e.g. =-=[58]-=-. Let G be any finite group. Let Φ be the set of all pairs (a, χ), where the a are representatives of the conjugacy classes of G and χ is the character of an irreducible representation of the centrali... |

235 |
Fusion rules and modular transformations in 2D conformal field theory
- Verlinde
- 1988
(Show Context)
Citation Context ...the rank of the VOA or the ‘central charge’ of the RCFT, and ha is the ‘conformal weight’ or L0-eigenvalue of the primary field a. Equation (2.1) is a special case of the so-called Verlinde’s formula =-=[83]-=-: V (g) a1 ···at = ∑ b∈Φ (S0b) 2(1−g)S a 1 b S0b · · · Sa t b S0b (3.7) It arose first in RCFT as an extremely useful expression for the dimensions of the space of conformal blocks on a genus g surfac... |

234 |
Classical and quantum conformal field theory
- Moore, Seiberg
- 1989
(Show Context)
Citation Context ...sentation the modular data of the RCFT. It has some interesting properties, as we shall see. Incidentally, there is in RCFT and related areas a (projective) representation of each mapping class group =-=[2,69,81]-=-. These groups play the role of modular group, for any Riemann surface with punctures. Their representations coming from e.g. RCFT are still poorly understood, and certainly deserve more attention, bu... |

215 |
Vertex operator algebras associated to representations of affine and Virasoro algebras
- Frenkel, Zhu
- 1992
(Show Context)
Citation Context ...-column of S consists of nonnegative real numbers (and also that the rank c is positive). In this context, Example 1 corresponds to the VOA associated to the lattice Λ [27]. Example 2 is recovered by =-=[37]-=-, who find a VOA structure on the highest weight X (1) r - module L(kΛ0); the other level k X (1) r -modules M = L(λ) all have the structure of VOA modules of V := L(kΛ0). Example 3 arises for example... |

140 |
Modular invariance of characters of vertex operator algebras
- Zhu
- 1996
(Show Context)
Citation Context ...ve eigenvector with eigenvalue 1: namely the eigenvector with ath component cha(i) (τ = i corresponds to q = e −2π > 0 and is fixed by τ ↦→ −1/τ; moreover the characters of VOAs converge at any τ ∈ H =-=[88]-=-). Unlike the S in (3.8a), the S of (3.8b) has no such eigenvector. We were lucky in our example that (3.8b) does obey MD1 ′ and even MD1 — typically for nonunitary RCFTs it won’t. The culprit is our ... |

130 |
H.: The operator algebra of orbifold models
- Dijkgraaf, Cumrun, et al.
- 1989
(Show Context)
Citation Context ...the centraliser CG(a). (Recall that the conjugacy class of an element a ∈ G consists of all elements of the form g −1 ag, and that the centraliser CG(a) is the set of all g ∈ G commuting with a.) Put =-=[25,67]-=- S (a,χ),(a ′ ,χ ′ ) = 1 ‖CG(a)‖ ‖CG(a ′ )‖ χ(a) T (a,χ),(a′,χ′) = δa,a ′δχ,χ′ χ(e) ∑ g∈G(a,a ′ ) χ ′ (g −1 ag)χ(ga ′ g −1 ) (3.6a) (3.6b) where G(a, a ′ ) = {g ∈ G | aga ′ g −1 = ga ′ g −1 a}, and e ... |

128 |
Borcherds, Vertex algebras, Kac-Moody algebras
- E
- 1986
(Show Context)
Citation Context ...ng to the affine algebra G (1) 2 level 1 contains the VOA L(28Λ0) =: L(0) corresponding to A (1) 1 at level 28. We get the branching rules L(Λ0) ′ = L(0) ⊕ L(10) ⊕ L(18) ⊕ L(28) and L(Λ2) ′ = L(6) ⊕ L=-=(12)-=- ⊕ L(16) ⊕ L(22). This corresponds to the A (1) 1 level 28 modular invariant given below in (6.1f). So knowing the modular invariants for some VOA V gives considerable information concerning its possi... |

127 |
Topological quantum field theories
- Atiyah
(Show Context)
Citation Context ...e of V ♮ , namely itself, and its character j(τ) − 744 is invariant under SL2(Z). Incidentally, there is in RCFT and related areas a (projective) representation of each mapping class group — see e.g. =-=[2,43,73,85]-=- and references therein. These groups play the role of modular group, for any Riemann surface. Their representations coming from e.g. RCFT are still poorly understood, and certainly deserve more atten... |

124 | Monstrous moonshine and monstrous Lie superalgebras, Invent
- Borcherds
- 1992
(Show Context)
Citation Context ...om this perspective, Monstrous Moonshine is maximally uninteresting — the corresponding representation is completely trivial! In fact, explaining that triviality was the hard part of Borcherds’ proof =-=[13]-=- of the Monstrous Moonshine conjectures. Let’s focus now on the former context. A rational conformal field theory ∗ (RCFT) has two vertex operator algebras (VOAs) V, V ′ . For simplicity we will take ... |

122 | Topological gauge theories and group cohomology
- Dijkgraaf, Witten
- 1990
(Show Context)
Citation Context ...ory explains how very general constructions (Goddard-Kent-Olive and orbifold) build up modular data from combinations of affine and finite group data — see e.g. [24]. This modular data can be twisted =-=[26]-=- by a 3-cocycle α ∈ H3 (G, C ×), which plays the same role here that level did in Example 2. A further major generalisation of this finite group data will be discussed in Example 6 below, and of this ... |

103 | Boundary conditions in rational conformal field theories
- Behrend, Pearce, et al.
(Show Context)
Citation Context ...correspond in RCFT to the 1-loop vacuum-to-vacuum amplitude Zαβ(t) of an open string, or equivalently of a cylinder whose edge circles are labelled by ‘conformally invariant boundary states’ |α〉, |β〉 =-=[15,76,41,10]-=-. In string theory these are called the ‘Chan-Paton degrees-of-freedom’ and are placed at the endpoints of open strings. The real variable −∞ < t < ∞ here is the modular parameter for the cylinder, an... |

102 |
Vertex algebras associated with even lattices
- Dong
- 1993
(Show Context)
Citation Context ...at, under this hypothesis, the 0-column of S consists of nonnegative real numbers (and also that the rank c is positive). In this context, Example 1 corresponds to the VOA associated to the lattice Λ =-=[27]-=-. Example 2 is recovered by [37], who find a VOA structure on the highest weight X (1) r - module L(kΛ0); the other level k X (1) r -modules M = L(λ) all have the structure of VOA modules of V := L(kΛ... |

97 | Modular{invariance of trace functions in orbifold theory
- Dong, Li, et al.
- 1997
(Show Context)
Citation Context ...modules M = L(λ) all have the structure of VOA modules of V := L(kΛ0). Example 3 arises for example in the orbifold of a self-dual lattice VOA by a subgroup G of the automorphism group of Λ (see e.g. =-=[30]-=-). An interpretation of fractional level affine algebra data should be possible along the lines of [29], who did it for A (1) 1 . Example 6: Subfactors. See e.g. [32,11] for good reviews of the subfac... |

94 |
Sewing constraints for conformal field theories on surfaces with boundaries
- Lewellen
- 1992
(Show Context)
Citation Context ...struction, generalising 6j-symbols to what are called Ocneanu cells, and extending the context to subparagroups. His new 15cells have been interpreted by [77] in terms of Moore-Seiberg-Lewellen data =-=[73,70]-=-. A very similar but simpler theory has been developed for type III factors. Bimodules now are equivalent to ‘sectors’, i.e. equivalence classes of endomorphisms λ : N → N (the corresponding subfactor... |

84 | A rational logarithmic conformal field theory Phys
- Gaberdiel, Kausch
- 1996
(Show Context)
Citation Context ...hey have a preferred basis Φ, unlike more familiar algebras. Incidentally, more general fusion-like rings arise naturally in subfactors (see Example 6 below) and nonrational logarithmic CFT (see e.g. =-=[45]-=-) so their theory also should be developed. Our treatment now will roughly follow that of Kawada’s C-algebras as given in [5]. The identity here is denoted by ‘1’ rather than the ‘0’ used in modular d... |

70 |
A proof of Verlinde’s formula
- Faltings
- 1994
(Show Context)
Citation Context ... choice of m ∈ {1, 2, 3}. Because of these isomorphisms, we know that the Nν λµ do indeed lie in Z≥, for any affine algebra. 10Also, they arise as dimensions of spaces of generalised theta functions =-=[33]-=-, as tensor product coefficients in quantum groups [42] and Hecke algebras [55] at roots of 1 and Chevalley groups for Fp [53], and in quantum cohomology [87]. For an explicit example, consider the si... |

64 | From Dynkin diagram symmetries to fixed point structures, preprint hep-th/9506135
- Fuchs, Schellekens, et al.
(Show Context)
Citation Context ...r instance, we are learning that the only finite ‘rational’ extensions of a generic affine VOA are those studied in [28] (‘simple-current extensions’) and whose modular data is conjecturally given in =-=[40]-=-. Another reason for studying modular invariants is that the answers are often surprising. Lists arising in math from complete classifications tend to be about as stale as 23phonebooks, but to give s... |

61 |
lattice integrable models associated with graphs, Nucl
- Francesco, Zuber, et al.
- 1990
(Show Context)
Citation Context ...elation between finite groups and Verlinde’s formula seems to have first been noticed in [68]. More generally, modular data is closely related to association schemes and C-algebras (as first noted in =-=[24]-=-, and independently in [4]), hypergroups [90], etc. That is to say, their axiomatic systems are similar. However, the exploration of an axiomatic system is influenced not merely by its intrinsic natur... |

60 |
Braid group statistics and their superselection rules
- Rehren
- 1990
(Show Context)
Citation Context ..., µ) ∈ Hom(λµ, µλ), which say roughly that λ and µ nearly commute (the ǫ ± must also obey some compatibility conditions, e.g. the Yang-Baxter equations). Once we have a nondegenerate braiding, Rehren =-=[74]-=- proved that we will automatically have modular data. We will return to subfactors in Section V. It would be very interesting in the subfactor picture to see to what the characters (1.1) correspond. F... |

59 |
Paths on Coxeter diagrams: From Platonic solids and singularities to minimal models and subfactors
- Ocneanu
(Show Context)
Citation Context ...ll be obeyed, except possibly commutativity: unfortunately in general A ⊗M B ̸∼ = B ⊗M A. We are interested in M and N being hyperfinite. An intricate subfactor invariant called a paragroup (see e.g. =-=[71,32]-=-) can be formulated in terms of 6j-symbols and fusion rings [32], and resembles exactly solvable lattice models in statistical mechanics. One way to get modular data is by passing from N ⊂ M to the as... |

52 |
An equivalence of fusion categories
- Finkelberg
- 1996
(Show Context)
Citation Context ...-known Littlewood-Richardson rule, for the affine fusions. Three preliminary steps in this direction are [78,80,34]. Identical numbers Nν λµ appear in several other contexts. For instance, Finkelberg =-=[35]-=- proved that the affine fusion ring is isomorphic to the K-ring of Kazhdan-Lusztig’s category Õ−k of level −k integrable highest weight X (1) r -modules, and to Gelfand-Kazhdan’s category Õq coming fr... |

50 |
The ADE classification of minimal and A (1)1 conformal invariant theories
- Cappelli, Itzykson, et al.
- 1987
(Show Context)
Citation Context ... Invariant Classifications The most famous modular invariant classification was the first. In (3.5) we gave explicitly the modular data for A (1) 1 level k. Its complete list of modular invariants is =-=[14]-=- (using the simple-current Ja = k − a) Ak+1 = D k 2 +2 = D k k∑ |χa| 2 , ∀k ≥ 1 (6.1a) a=0 k∑ a=0 χa χJ a a , whenever k 2 2 +2 = |χ0 + χJ0| 2 + |χ2 + χJ2| 2 + · · · + 2|χ k 2 is odd (6.1b) | 2 , when... |

50 |
Branes: from free fields to general backgrounds
- Fuchs, Schweigert
- 1998
(Show Context)
Citation Context ...correspond in RCFT to the 1-loop vacuum-to-vacuum amplitude Zαβ(t) of an open string, or equivalently of a cylinder whose edge circles are labelled by ‘conformally invariant boundary states’ |α〉, |β〉 =-=[15,76,41,10]-=-. In string theory these are called the ‘Chan-Paton degrees-of-freedom’ and are placed at the endpoints of open strings. The real variable −∞ < t < ∞ here is the modular parameter for the cylinder, an... |

47 | The many faces of Ocneanu cells
- Petkova, Zuber
(Show Context)
Citation Context ...], Ocneanu has significantly refined this construction, generalising 6j-symbols to what are called Ocneanu cells, and extending the context to subparagroups. His new 15cells have been interpreted by =-=[77]-=- in terms of Moore-Seiberg-Lewellen data [73,70]. A very similar but simpler theory has been developed for type III factors. Bimodules now are equivalent to ‘sectors’, i.e. equivalence classes of endo... |

44 |
Littlewood-Richardson Coefficients for Hecke Algebras at
- Goodman, Wenzl
- 1990
(Show Context)
Citation Context ... do indeed lie in Z≥, for any affine algebra. 10Also, they arise as dimensions of spaces of generalised theta functions [33], as tensor product coefficients in quantum groups [42] and Hecke algebras =-=[55]-=- at roots of 1 and Chevalley groups for Fp [53], and in quantum cohomology [87]. For an explicit example, consider the simplest affine algebra (A (1) 1 ) at level k. We may take P k + = {0, 1, . . ., ... |

39 |
A classical invitation to algebraic numbers and class fields
- Cohn
- 1978
(Show Context)
Citation Context ...e definition of fusion ring or modular data. Here is an intriguing example, inspired by (4.4) below. Example 7 [49]: Number fields. A basic introduction to algebraic number theory is provided by e.g. =-=[17]-=-. Choose any finite normal extension L of Q, and find any totally positive α ∈ L with Tr(|α| 2 ) = 1 (total positivity will turn out to be necessary for F1). Now find any Qbasis x1 = 1, x2, . . ., xn ... |

37 |
Modular invariant representations of infinitedimensional Lie algebras and superalgebras
- Kac, Wakimoto
- 1988
(Show Context)
Citation Context ...nd their formulas are also surprisingly compact. Incidentally, an analogous modular transformation matrix S to (3.2b) exists for the socalled admissible representations of X (1) r at fractional level =-=[62]-=-. The matrix is symmetric, but has no column of constant phase and thus naively putting it into Verlinde’s formula (2.1) will necessarily produce some negative numbers (apparently they’ll always be in... |

31 |
Simple currents and extensions of vertex operator algebras
- Dong, Li, et al.
- 1996
(Show Context)
Citation Context ... V gives considerable information concerning its possible ‘nice’ extensions V ′ . For instance, we are learning that the only finite ‘rational’ extensions of a generic affine VOA are those studied in =-=[28]-=- (‘simple-current extensions’) and whose modular data is conjecturally given in [40]. Another reason for studying modular invariants is that the answers are often surprising. Lists arising in math fro... |

28 |
Quantum Invariants of Knots and
- Turaev
- 1994
(Show Context)
Citation Context ...sentation the modular data of the RCFT. It has some interesting properties, as we shall see. Incidentally, there is in RCFT and related areas a (projective) representation of each mapping class group =-=[2,69,81]-=-. These groups play the role of modular group, for any Riemann surface with punctures. Their representations coming from e.g. RCFT are still poorly understood, and certainly deserve more attention, bu... |

28 |
Operator algebras and conformal field theory
- Wassermann
- 1998
(Show Context)
Citation Context ...8b)). Jones and Wassermann have explicitly constructed the affine algebra subfactors of Example 2, at least for A (1) r , and Wassermann later showed they recover the affine algebra fusions (see e.g. =-=[85]-=-). To any subgroup-group pair H < G, we can obtain a subfactor 15R×H ⊂ R×G of crossed products, where R is the type II1 hyperfinite factor, and thus a (not necessarily commutative) fusion-like ring [... |

28 |
Fusion rules and logarithmic representations of a WZW model at fractional level, hep-th/0105046
- Gaberdiel
(Show Context)
Citation Context ...roduct of the A1,p−2 fusion ring with a fusion ring 11at ‘level’ q − 1 associated to the rank 1 supersymmetric algebra osp(1|2). Some doubt however on the relevance of these efforts has been cast by =-=[46]-=-. A similar theory should exist at least for the other A (1) r ; initial steps for A (1) 2 have been made in [45]. Exactly how these correspond to modular data, or rather how modular data should be ge... |

27 | Boundary conformal field theory and fusion ring representations
- Gannon
- 2002
(Show Context)
Citation Context ...y is equivalent to demanding that the identity 0 occurs in E(N) with multiplicity 1. We are interested in irreducible equivalence classes of NIM-reps. It can be shown there will be only finitely many =-=[50]-=-. Two useful facts are: the Perron-Frobenius eigenvalue of Na is the q-dimension Sa0 S00 (we’ll see this used next section); and for all a ∈ Φ, ∑ Sab S0b b∈E = Tr(Na) ∈ Z≥ (5.6) The consequences of th... |

20 |
Fusion rules for the fractional level ̂sl(2) algebra
- Awata, Yamada
- 1185
(Show Context)
Citation Context ....1) will necessarily produce some negative numbers (apparently they’ll always be integers though). A legitimate fusion ring has been obtained for A (1) 1 at fractional level k = p q − 2 in other ways =-=[3]-=-; it factorises into the product of the A1,p−2 fusion ring with a fusion ring at ‘level’ q − 1 associated to the rank 1 supersymmetric algebra osp(1|2). A similar theory should exist at least for the ... |

20 |
Fusion rules in conformal field theory, Fortschr
- Fuchs
- 1994
(Show Context)
Citation Context ...e somewhat more terse. Example 4: RCFT, TFT. See e.g. [24] and [81], and references therein, for good surveys of 2-dimensional conformal and 3-dimensional topological field theories, respectively. In =-=[38]-=- can be found a survey of fusion rings in rational conformal field theory (RCFT). As discussed earlier, a major source of modular data comes from RCFT (and string theory) and, more or less the same th... |

19 | Vertex operator algebras associated to admissible representations of ŝl2
- Dong, Li, et al.
- 1997
(Show Context)
Citation Context ...the orbifold of a self-dual lattice VOA by a subgroup G of the automorphism group of Λ (see e.g. [30]). An interpretation of fractional level affine algebra data should be possible along the lines of =-=[29]-=-, who did it for A (1) 1 . Example 6: Subfactors. See e.g. [32,11] for good reviews of the subfactor ↔ CFT relation. The final general source of modular data which we will discuss comes from subfactor... |

18 | Errata: ”Algorithm for WZW fusion rules: a proof”, Phys - Walton - 1990 |

18 | On the TQFT representations of the mapping class groups
- Funar
(Show Context)
Citation Context ...e of V ♮ , namely itself, and its character j(τ) − 744 is invariant under SL2(Z). Incidentally, there is in RCFT and related areas a (projective) representation of each mapping class group — see e.g. =-=[2,43,73,85]-=- and references therein. These groups play the role of modular group, for any Riemann surface. Their representations coming from e.g. RCFT are still poorly understood, and certainly deserve more atten... |

16 |
The kernel of the modular representation and the Galois action
- Bantay
(Show Context)
Citation Context ... [16] (and that proof assumes additional axioms)! In fact we still don’t have a finiteness theorem: for a given cardinality ‖Φ‖, prove that there are only finitely many possible modular data. But see =-=[31,6]-=- for more sophisticated and promising approaches to modular data classification. The matrix T is fairly poorly constrained by MD1–MD4. Another axiom, obeyed by all the main examples next section, can ... |

16 |
Geometric aspects of quantum field theory
- Segal
- 1990
(Show Context)
Citation Context ...ader a sense of the context. Detailed proofs will appear elsewhere. The theory is still a work-in-progress, and emphasis is given here to several open questions and problems. In Segal’s axioms of CFT =-=[79]-=-, any Riemann surface with boundary is assigned a certain linear homomorphism. Roughly speaking, Borcherds [12] axiomatised this data corresponding to a sphere with 3 disks removed, and the result is ... |

15 |
Modular invariants from simple currents: an explicit proof,” Phys
- Schellekens, Yankielowicz
- 1989
(Show Context)
Citation Context ... ≤ ‖Φ‖ m−1 Tr(N km a ) (4.2d) The inequality (4.2b) suggests that we look at those primaries a ∈ Φ obeying the equality Sa0 = S00. Such primaries are called simple-currents in RCFT parlance (see e.g. =-=[77,24]-=- and references therein), but the much more obvious mathematical name is units. 18To any unit j ∈ Φ, there is a phase ϕj : Φ → C and a permutation J of Φ such that j = J0 and SJa,b = ϕj(b) Sa,b TJa,J... |

14 |
Affine Lie algebras, weight multiplicities and branching rules, Vol
- Kass, Moody, et al.
- 1990
(Show Context)
Citation Context ...1 index-3 subgroup of SL2(Z). Nothing essential though is lost, so the definition of modular data should be broadened to include all integral lattice examples. Example 2: Kac-Moody algebras. See e.g. =-=[60,63]-=- for the basics of Kac-Moody algebras. The source of some of the most interesting modular data are the affine nontwisted KacMoody algebras X (1) r . The simplest way to construct affine algebras is to... |

13 |
Lattice integrable models and modular invariance
- Francesco, Zuber, et al.
- 1989
(Show Context)
Citation Context ...elation between finite groups and Verlinde’s formula seems to have first been noticed in [65]. More generally, modular data is closely related to association schemes and C-algebras (as first noted in =-=[23]-=-, and independently in [4]), hypergroups [86], etc. That is to say, their axiomatic systems are similar. However, the exploration of an axiomatic system is influenced not merely by its intrinsic natur... |

12 | On the classification of modular fusion algebras
- Eholzer
- 1995
(Show Context)
Citation Context ... [16] (and that proof assumes additional axioms)! In fact we still don’t have a finiteness theorem: for a given cardinality ‖Φ‖, prove that there are only finitely many possible modular data. But see =-=[31,6]-=- for more sophisticated and promising approaches to modular data classification. The matrix T is fairly poorly constrained by MD1–MD4. Another axiom, obeyed by all the main examples next section, can ... |

10 | Berenstein-Zelevinski triangles, elementary coupling and fusion rules
- Begin, Kirillov, et al.
- 1993
(Show Context)
Citation Context ... 4 N c ab = { 1 if c ≡ a+b (mod 2) and |a−b| ≤ c ≤ min{a+b, 2k−a−b} (3.5c) 0 otherwise The only other affine algebras for which the fusions have been explicitly calculated are A (1) 2 [8] and A (1) 3 =-=[9]-=-, and their formulas are also surprisingly compact. Incidentally, an analogous modular transformation matrix S to (3.2b) exists for the socalled admissible representations of X (1) r at fractional lev... |

9 |
Exotic Fourier transform
- Lusztig
- 1994
(Show Context)
Citation Context ...the centraliser CG(a). (Recall that the conjugacy class of an element a ∈ G consists of all elements of the form g −1 ag, and that the centraliser CG(a) is the set of all g ∈ G commuting with a.) Put =-=[25,67]-=- S (a,χ),(a ′ ,χ ′ ) = 1 ‖CG(a)‖ ‖CG(a ′ )‖ χ(a) T (a,χ),(a′,χ′) = δa,a ′δχ,χ′ χ(e) ∑ g∈G(a,a ′ ) χ ′ (g −1 ag)χ(ga ′ g −1 ) (3.6a) (3.6b) where G(a, a ′ ) = {g ∈ G | aga ′ g −1 = ga ′ g −1 a}, and e ... |