By Urmie Ray
A central element within the evidence of the Moonshine Theorem, connecting the Monster team to modular varieties, is the countless dimensional Lie algebra of actual states of a chiral string on an orbifold of a 26 dimensional torus, known as the Monster Lie algebra. it's a Borcherds-Kac-Moody Lie algebra with Lorentzian root lattice; and has an linked automorphic shape having a product growth describing its constitution. Lie superalgebras are generalizations of Lie algebras, helpful for depicting supersymmetry – the symmetry concerning fermions and bosons. such a lot recognized examples of Lie superalgebras with a comparable automorphic shape similar to the pretend Monster Lie algebra whose mirrored image staff is given via the Leech lattice come up from (super)string thought and will be derived from lattice vertex algebras. The No-Ghost Theorem from twin resonance concept and a conjecture of Berger-Li-Sarnak at the eigenvalues of the hyperbolic Laplacian supply robust proof that they're of rank at so much 26.
The target of this publication is to offer the reader the instruments to appreciate the continuing category and development venture of this category of Lie superalgebras and is perfect for a graduate path. the mandatory history is given inside chapters or in appendices.
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Additional info for Automorphic Forms and Lie Superalgebras
Hint: Consider aﬃne BKM superalgebras. 5. )0 is positive deﬁnite on G. Show that the BKM algebra G is an aﬃne Lie algebra if and only if it is semi-positive deﬁnite on G. 3 The Root System The generalized Cartan subalgebra H acts semisimply on the BKM superalgebra G via the adjoint action. Hence to know the structure of G, it is worth ﬁnding its eigenvalues and eigenspaces. It is most natural to take the eigenvalues in the dual space H ∗ . In order to do this without any complications, the elements hi are assumed in [Kac14] to be linearly independent.
6. Let G be a BKM algebra. e. ([g, x], y)0 = −(x, [ω0 g, y])0 for all g, x, y ∈ G. Proof. )0 is Hermitian. The contravariance property is left for the reader to check. 29. Let x = x1 + ix2 ∈ G and y = y1 + iy2 ∈ G, where xi , yi ∈ C(A) for i = 1, 2. Then, (y, x)0 = (y1 − iy2 , x1 + ix2 ) = (x1 − ix2 , y1 + iy2 ) = (x, y)0 . )0 to be Hermitian, the map ω0 has to be assumed to be antilinear. )0 for ﬁnite dimensional semisimple Lie algebras is classical. 6. 7. )0 is positive deﬁnite on G. This result is clearly false in this strong form for arbitrary BKM algebras since the generalized symmetric Cartan matrix A may have non-positive diagonal entries.
4. ) on the BKM superalgebra G. e. equal to its derived Lie subalgebra), then this form is unique up to multiplication by a non-zero scalar. ) will denote a non-degenerate, invariant, supersymmetric, consistent bilinear form on G. 1, for ﬁnite dimensional BKM superalgebras of type D(2, 1, α), the non-degenerate forms are not multiples of the Killing form. However these properties of a bilinear form are too weak to characterize BKM algebras, let alone BKM superalgebras. 2). To ﬁnd a characterization of BKM algebras via bilinear forms, we need to consider the compact antilinear automorphism.