Realizations of the Monster Lie algebra

Jurisich, Elizabeth; Lepowsky, James; Wilson, Robert Lee

doi:10.1007/bf01614075

Cited by 30 publications

(27 citation statements)

References 24 publications

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“…[18], and also [146]), which, in turn, implies the replication formulas formulated in [58,191]. Taking g = e in (3.16) we recover (3.9), so (3.16) furnishes a natural, monsterindexed family of analogues of the identity (3.9).…”

Section: Theorem 2 (Frenkel-lepowsky-meurman)mentioning

confidence: 68%

“…Define m i (−m, n) to be the multiplicity of Proof. The claim follows from the modification of Borcherds' proof of Theorem 4 presented by Jurisich-Lepowsky-Wilson in [146]. In [146] a certain free Lie sub algebra u − of the monster Lie algebra m is identified, for which the identity (u − ) = H(u − ) (or rather, the logarithm of this) yields The coefficient of p m q n in the right-hand side of (7.17) is evidently a non-negative integer combination of the M-modules V n , so the proof of the claim is complete.…”

Section: ) In Particular T (−M) G (τ ) Is a Monic Integral Polynommentioning

confidence: 94%

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Moonshine

Duncan

Griffin

Ono

2015

Mathematical Sciences

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illustrate the case of J(τ ) = j(τ ) − 744, whose coefficients turn out to be sums of the dimensions of the 194 irreducible representations of M. Such formulas are dictated by the structure of the graded monstrous moonshine modules. Recent works in moonshine suggest deep relations between number theory and physics. Number theoretic Kloosterman sums have reappeared in quantum gravity, and mock modular forms have emerged as candidates for the computation of black hole degeneracies. This paper is a survey of past and present research on moonshine. We also compute the quantum dimensions of the monster orbifold and obtain exact formulas for the multiplicities of the irreducible components of the moonshine modules. These formulas imply that such multiplicities are asymptotically proportional to dimensions.

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Section: Theorem 2 (Frenkel-lepowsky-meurman)mentioning

confidence: 68%

Section: ) In Particular T (−M) G (τ ) Is a Monic Integral Polynommentioning

confidence: 94%

Moonshine

Duncan

Griffin

Ono

2015

Mathematical Sciences

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“…There are a variety of deÿnitions in the literature: See for example [1][2][3][4], [9][10][11]13]. We will work in the setting of [9][10][11][12], where the construction is compatible with the deÿnition of a Lie algebra from a deÿning matrix given in [13]. The following is a summary of the notation and results of [10] for Borcherds algebras (called generalized Kac-Moody algebras in [10]).…”

Section: Borcherds Lie Algebrasmentioning

confidence: 98%

“…[12]). The monster Lie algebra appears in the study of the correspondence between graded characters of the Monster simple group and modular functions, i.e.…”

Section: Introductionmentioning

confidence: 93%

A resolution for standard modules of Borcherds Lie algebras

Jurisich

2004

Journal of Pure and Applied Algebra

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In this paper we construct a Bernstein-Gelfand-Gelfand-type resolution of a standard (irreducible) module for a Borcherds Lie algebra g. The resolution involves generalized Verma modules; the "reductive part" of the "parabolic subalgebra" of g may be a Borcherds algebra. This resolution, which generalizes that of Garland and Lepowsky for Kac-Moody algebras, is constructed here in su cient generality to include the cases of the monster and fake monster Lie algebras. These algebras are of particular interest because of their connection to vertex algebras and conformal ÿeld theory. The monster Lie algebra is of importance because it is constructed from the moonshine module vertex algebra, appearing in the study of "monstrous moonshine" relating to the Fischer-Griess Monster simple group. The resolution constructed in this paper is used to obtain Kostant's homology formula for Borcherds algebras, and the character formula for a standard module.

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“…m inherits a II 1,1 -grading from V 1,1 , and this is its root space decomposition: the (m, n) root space is isomorphic (as a vector space) to V m n , if (m, n) = (0, 0); the (0, 0) piece is isomorphic to R 2 . Structurally, the Monster Lie algebra has a decomposition m = u + ⊕ gl 2 ⊕ u − into a sum of Lie subalgebras, where u ± are infinitely generated free Lie algebras (see, for example, [71]). It inherits the action of M from V .…”

Section: The Monster Lie Algebra Mmentioning

confidence: 99%