Generalized Kac-Moody algebras and some related topics

Abstract

To the memory of my father IntroductionIn this review article, we shall give an introduction to the field of generalized KacMoody algebras. This subject gained much interest when Borcherds first proved the remarkable Moonshine Conjecture, which connects two areas apparently far apart: on the one hand, the Monster simple group and on the other elliptic modular functions. Any connection found between an object which has as yet played a limited abstract role and a more fundamental concept is always very fascinating. Also it is not surprising that ideas on which the proof of such a result are based would give rise to many new questions, thus opening up different research directions and finding applications in a wide variety of areas.We shall start by explaining the Moonshine Conjecture, giving at the same time a brief history of it. We will then give a brief outline of its proof. In the following chapters, we will take one by one the important concepts used in the proof (generalized Kac-Moody algebras, Vertex algebras, automorphic forms), explain them in some detail and mention recent developments connected to them. We shall finish by briefly discussing two interesting topics related to the field of generalized Kac-Moody algebras: Lie superalgebras and lattices.The author would like to offer her apologies to anyone whose contribution was omitted or not extensively covered. This is not intentional.Remark. Throughout this paper, the base field will be C. ContentsThe modular invariant j is a complex valued holomorphic function on the upper half plane H = {x + iy|y > 0}, which is fixed by the discrete group SL(2, Z):for all τ ∈ H, and g ∈ SL(2, Z), j(gτ ) = j(τ ). GENERALIZED KAC-MOODY ALGEBRAS AND SOME RELATED TOPICS 3We remind the reader that the group Γ = SL(2, Z) acts on H in the following way.Let a, b, c, d ∈ Z be integers such that g = a b c d ∈ Γ. Then for any point τ ∈ H,The modular invariant j is meromorphic at infinity and is a bijection from the compactification of H/Γ, H ∪ {i∞}/Γ to the compact Riemann sphere C ∪ {∞} such that j(i∞) = ∞. Hence any meromorphic function on H fixed by Γ is a rational function of j. Since j(τ + 1) = j(τ ), we can write j as a function of q = e 2πiτ . Since J is holomorphic on H, as a function of q, J has a Laurent series in the punctured disc of radius 1 centered at 0: would be an interesting function. When c(n) was a known linear combination, the M -modules V (n) were to be taken to be the sums of the irreducible representations whose dimensions appeared in the linear combination. As in the case of V (1), the head representations are obviously not just sums of the trivial module! Thus their work suggested these series, now known as the McKay-Thompson series, were worth investigating. For g = 1, as already noted, we get the function J. Furthermore, they conjectured that M has a natural infinite dimensional representationwith graded character T g . It came as a surprise that the most natural module of a finite group was infinite dimensional. Moonshine Conjecture. For all g ∈ M ,...

Moonshine

“…Borcherds generalized this further, allowing also copies of the three-dimensional Heisenberg Lie algebra to serve as building blocks, and thus arrived [16] at the notion of generalized Kac-Moody algebra, or Borcherds-Kac-Moody (BKM) algebra, which has subsequently found many applications in mathematics and mathematical physics (cf. [138,206]). …”

On classification of Lorentzian Kac-Moody algebras

“…There are many papers and reviews written on the subject of Borcherds solution [5] of Moonshine Conjecture. E. g. see [6], [10], [27], [70]. We don't consider that in the paper.…”

On Heisenberg-like Super Group Structures

“…First of all, it is a Lie superalgebra, and therefore [Πu, Πv] is really an anticommutator. There are some others generalizations in the literature (e.g., [1,6,7] or [9]). The main difference is that our construction introduces, not a 1-dimensional, but a (1, 1)-dimensional center; besides, the bilinear formB does not need to have a specific type of symmetry or skew-symmetry, nor need it be non-degenerate.…”