Sphere Packings, Lattices and Groups

Introduction to lie algebras and representation theory

1. (i) Suppose K is a conjugacy class of Sn contained in An; then K is called split if K is a union of two conjugacy classes of An. Show that the number of split conjugacy classes contained in An is equal to the number of characters χ ∈ Irr(Sn) such that χAn is not irreducible. (Hint. Consider the vector space of class functions on An which are invariant under conjugation by the transposition (12).)

/pdf/character-theory-of-finite-groups-mraeb8b123.pdf

Character Theory of Finite Groups

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/63B506EB85D6875966CE63BA0336BC2F/S0025557200190515a.pdf/div-class-title-span-class-italic-lie-groups-and-lie-algebras-chapters-1-3-span-by-n-bourbaki-pp-450-dm-98-1989-isbn-3-540-50218-1-springer-div.pdf

Lie groups and Lie algebras (chapters 1-3 ), by N. Bourbaki. Pp 450. DM 98. 1989. ISBN 3-540-50218-1 (Springer)

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Reflection Groups And Coxeter Groups

http://www.ncts.ncku.edu.tw/math/files/paper/2009-01-04.pdf

W -algebras related to parafermion algebras

The structure of parafermion vertex operator algebras

Zhu's algebra, C2-algebra and C2-cofiniteness of parafermion vertex operator algebras

This paper is a continuation of (33) at which several coset subalgebras of the lattice VOA Vp 2E8 were constructed and the relationship between such algebras with the famous McKay observation on the extended E8 diagram and the Monster simple group were discussed In this article, we shall provide the technical details We completely determine the structure of the coset subalgebras constructed and show that they are all generated by two conformal vectors of central charge 1/2 We also study the represen- tation theory of these coset subalgebras and show that the product of two Miyamoto involutions is in the desired conjugacy class of the Monster simple group if a coset subal- gebra U is actually contained in the Moonshine VOA V ♮  The existence of U inside the Moonshine VOA V ♮ for the cases of 1A,2A,2B and 4A is also established Moreover, the cases for 3A, 5A and 3C are discussed and the Monster simple group were discussed In this article, we shall provide the technical details We shall determine the structure of the coset subalgebras and show that they are all generated by two conformal vectors of central charge 1/2 We also study the representation theory of these coset subalgebras and show that the product of two Miyamoto involutions is in the desired conjugacy class of the Monster simple group if a coset subalgebra U is actually contained in the Moonshine

Mckay's observation and vertex operator algebras generated by two conformal vectors of central charge 1/2

In this paper, we study the structure of a general framed vertex operator algebra (VOA). We show that the structure codes (C,D) of a framed VOA V satisfy certain duality conditions. As a consequence, we prove that every framed VOA is a simple current extension of the associated binary code VOA V
 C
 . This result suggests the feasibility of classifying framed vertex operator algebras, at least if the central charge is small. In addition, the pointwise frame stabilizer of V is studied. We completely determine all automorphisms in the pointwise stabilizer, which are of order 1, 2 or 4. The 4A-twisted sector and the 4A-twisted orbifold theory of the famous moonshine VOA $$V^\natural$$
 are also constructed explicitly. We verify that the top module of this twisted sector is of dimension 1 and of weight 3/4 and the VOA obtained by 4A-twisted orbifold construction of $$V^\natural$$
 is isomorphic to $$V^\natural$$
 itself.

/pdf/on-the-structure-of-framed-vertex-operator-algebras-and-5f5vyuyad2.pdf

Ching Hung Lam

Papers

W -algebras related to parafermion algebras

The structure of parafermion vertex operator algebras

Zhu's algebra, C2-algebra and C2-cofiniteness of parafermion vertex operator algebras

Mckay's observation and vertex operator algebras generated by two conformal vectors of central charge 1/2

On the Structure of Framed Vertex Operator Algebras and Their Pointwise Frame Stabilizers