### Citations

1317 |
Categories for the Working Mathematician.
- Lane
- 1997
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Citation Context ...gs to the axioms of higher dimensional categories with a tensor product. This nonassociativity disappears in low dimensional field theories, for which the special modular tensor categories are strict =-=[1]-=-. A non trivial braiding exists for a generic representation category. Starting with the classification of complex Lie algebras, we see how the indefinite metrics of spacetime are associated to except... |

1055 |
Infinite-dimensional Lie algebras
- Kac
- 1985
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Citation Context ..., q, r ≥ 2, where we assume that p ≥ q ≥ r. Allowing also length 1 lines, the ternary graphs include the SU(N) graphs of type (N − 1, 1, 1). The single node is the (1, 1, 1) graph. A useful number is =-=[23]-=- D(p, q, r) ≡ 1 p + 1 q + 1 r . (8) Lemma 2.4. For a ternary graph, the determinant of A equals pqr(D(p, q, r)− 1). Proof: For the SU(N) matrices, the determinant N = p + q follows by induction on N .... |

579 |
Sphere Packings, Lattices and Groups
- Conway, Sloane
- 1993
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Citation Context ...r discriminant ∆ ≡ ∑ n τ(n)qn = 2403E43 − 5042E62 123 = q−24q2+252q3−1472q4+· · · (13) Then θ12 ≡ (240E4)3 − 720∆ = 1 + 196560q2 + 16773120q3 + · · · (14) counts the root vectors in the Leech lattice =-=[25]-=-[26]. In this case, the minimal vectors have norm 4. Using the complex form of the SU(3) lattice, we will think of E8 as a lattice in C4 and the Leech lattice as a lattice in C12. 7 Definition 3.1 An ... |

199 |
The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series,
- Ono
- 2004
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Citation Context ...d using octonion triplets [22]. As in (14), the vectors of norm 2n in LL [27] are counted by θ12 = 240(33600 · E43 + 441 · E62). (27) The form E12 is not independent of E4 and E6. For instance, using =-=[30]-=- 3∆ = 65 252 E12 − 691 · E62, (28) we have E12 = 252 · 50 13 (96 · E43 +E62) (29) and θ12 = 240( 1 3 E12 + 84 · E62). (30) 10 4 Noncommutativity and Nonassociativity Let ω be a complex number. In cate... |

87 |
Lectures on Exceptional Lie Groups
- Adams
- 1996
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Citation Context ...y even, unimodular lattice of rank 8 is the E8 lattice. The SO(16) lattice is the even sublattice of the simple square lattice in Rc, but its determinant equals 4. Let us now build the E8 lattice [27]=-=[28]-=-. Let α and β be the simple roots of the SU(3) lattice in C, as in (2), and ν the dual basis vector ν ≡ α− β√ 3 = 1√ 6 + 1√ 2 i (15) of length 2/3. There are four SU(3) directions in C4, and a basis f... |

80 | A Taste of Jordan Algebras, - McCrimmon - 2004 |

40 |
Jr, Lectures on tensor categories and modular functors
- Bakalov, Kirrilov
- 2001
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Citation Context ...basis for R3, with the triangle representing the origin. Morally, space begins with affine Dynkin diagrams. 2 The duality between points and planes underlies the diagrams of a modular tensor category =-=[13]-=-[14], which is a category with a braiding and ribbon twists. The ambient plane stands for a single zero dimensional object. A braid strand is a one dimensional object, going from the left hand side of... |

32 |
Quantum Groups: a Path to Current Algebra.
- Street
- 2007
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Citation Context ...nd 3-arrows form a ⊗ category, and there exist braiding arrows γXY : X ⊗ Y → Y ⊗X (38) for every pair of 2-arrows X and Y . A braiding generalises anticommutativity. These γXY obey the hexagon axioms =-=[31]-=-, aXY Z · γX(Y Z) · aY ZX (39) = (γXY ⊗ 1Z) · aY XZ · (1Y ⊗ γXZ), and similarly for a−1. We ignore the additional left and right unit maps [31]. Homework: draw these axioms with objects. A braided ⊗ c... |

30 |
Twelve Sporadic Groups
- Griess
- 1998
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Citation Context ... only even, unimodular lattice of rank 8 is the E8 lattice. The SO(16) lattice is the even sublattice of the simple square lattice in Rc, but its determinant equals 4. Let us now build the E8 lattice =-=[27]-=-[28]. Let α and β be the simple roots of the SU(3) lattice in C, as in (2), and ν the dual basis vector ν ≡ α− β√ 3 = 1√ 6 + 1√ 2 i (15) of length 2/3. There are four SU(3) directions in C4, and a bas... |

21 | Modular forms, a computational approach, - Stein - 2007 |

2 |
Lectures on quantum groups
- Du
- 1994
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Citation Context ...ns emerge from the axiomatic structure of quantum representation categories, without resorting to a Lagrangian formalism. For affine algebras, one considers deformation parameters t 6= 0,±1 for Ut(g) =-=[18]-=-[19][20], the quantum version of the universal enveloping algebra U(g). In a nice modular tensor category, the value of t is entirely determined by finiteness conditions on the representations. Beyond... |

1 |
Acta Phys. Polonica B9
- Ruegg
- 1978
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Citation Context ...structure. The octonions are introduced in the context of affine algebras. 1 Overview It has long been considered that color SU(3) and quark confinement depend on the nonassociative octonions O [2][3]=-=[4]-=-[5][6][7][8]. The complex exceptional Lie algebras are all related to the octonions O. For instance, in [9][10][11] one obtains a real form of the group E6 as the group SL(3,O), the 3× 3 matrices of d... |

1 |
lectures on Conformal Field Theory at ANU
- Itzykson
- 1994
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Citation Context ...the maximal torus defined by the root lattice. For an affine algebra, h defines a c + 2 dimensional (over C) subalgebra (1⊗ h)⊕ CK ⊕ C(−L0) (49) in g(1). The value c is chosen here by the formula [23]=-=[32]-=-[33] c(k) = dimg · k k + h∨ (50) for the standard representation category, where we only consider k = 1. For E8(1), h∨ = 30 and c = 8. For E (1) 6 , h ∨ = 12 and c = 6. In the ADE case, the dual Coxet... |