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Automorphic forms and Lorentzian KacMoody algebras
 Part II,, Preprint RIMS 1122, Kyoto
, 1996
"... Abstract. Using the general method which was applied to prove finiteness of the set of hyperbolic generalized Cartan matrices of elliptic and parabolic type, we classify all symmetric (and twisted to symmetric) hyperbolic generalized Cartan matrices of elliptic type and of rank 3 with a lattice Weyl ..."
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Cited by 39 (19 self)
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Abstract. Using the general method which was applied to prove finiteness of the set of hyperbolic generalized Cartan matrices of elliptic and parabolic type, we classify all symmetric (and twisted to symmetric) hyperbolic generalized Cartan matrices of elliptic type and of rank 3 with a lattice Weyl vector. We develop the general theory of reflective lattices T with 2 negative squares and reflective automorphic forms on homogeneous domains of type IV defined by T. We consider this theory as mirror symmetric to the theory of elliptic and parabolic hyperbolic reflection groups and corresponding hyperbolic root systems. We formulate Arithmetic Mirror Symmetry Conjecture relating both these theories and prove some statements to support this Conjecture. This subject is connected with automorphic correction of Lorentzian Kac–Moody algebras. We define Lie reflective automorphic forms which are the most beautiful automorphic forms defining automorphic Lorentzian Kac–Moody algebras and formulate finiteness Conjecture for these forms. Detailed study of automorphic correction and Lie reflective automorphic forms for generalized Cartan matrices mentioned above will be given in Part II. 0.
The arithmetic mirror symmetry and Calabi–Yau manifolds
 Preprint RIMS Kyoto Univ. RIMS1129
, 1997
"... Abstract. We extend our variant of mirror symmetry for K3 surfaces [GN3] and clarify its relation with mirror symmetry for Calabi–Yau manifolds. We introduce two classes (for the models A and B) of Calabi–Yau manifolds fibrated by K3 surfaces with some special Picard lattices. These two classes are ..."
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Cited by 12 (10 self)
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Abstract. We extend our variant of mirror symmetry for K3 surfaces [GN3] and clarify its relation with mirror symmetry for Calabi–Yau manifolds. We introduce two classes (for the models A and B) of Calabi–Yau manifolds fibrated by K3 surfaces with some special Picard lattices. These two classes are related with automorphic forms on IV type domains which we studied in our papers [GN1]—[GN6]. Conjecturally these automorphic forms take part in the quantum intersection pairing for model A, Yukawa coupling for model B and mirror symmetry between these two classes of Calabi–Yau manifolds. Recently there were several papers by physicists where it was shown on some examples. We propose a problem of classification of introduced Calabi–Yau manifolds. Our papers [GN1]—[GN6] and [N3]—[N14] give a hope that this is possible. They describe possible Picard or transcendental lattices of general K3 fibers of the Calabi–Yau manifolds. In [GN3] we suggested a variant of mirror symmetry for K3 surfaces which is related with reflection groups in hyperbolic spaces and automorphic forms on IV
On classification of Lorentzian KacMoody algebras
, 2002
"... We discuss a general theory of Lorentzian Kac–Moody algebras which should be a hyperbolic analogy of the classical theories of finitedimensional semisimple and affine Kac–Moody algebras. First examples of Lorentzian Kac–Moody algebras were found by Borcherds. We consider general finiteness results ..."
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Cited by 12 (5 self)
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We discuss a general theory of Lorentzian Kac–Moody algebras which should be a hyperbolic analogy of the classical theories of finitedimensional semisimple and affine Kac–Moody algebras. First examples of Lorentzian Kac–Moody algebras were found by Borcherds. We consider general finiteness results about the set of Lorentzian Kac–Moody algebras and the problem of their classification. As an example, we give classification of Lorentzian Kac–Moody algebras of the rank three with the hyperbolic root lattice S ∗ t, symmetry lattice L∗t, and the symmetry group Ô +(Lt), t ∈ N, where
On the classification of hyperbolic root systems of the rank three, Part III
, 1999
"... See Parts I and II in alggeom/9711032 and 9712033. Here we classify maximal hyperbolic root systems of the rank three having restricted arithmetic type and a generalized lattice Weyl vector ρ with ρ 2 < 0 (i. e. of the hyperbolic type). We give classification of all reflective of hyperbolic typ ..."
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Cited by 6 (3 self)
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See Parts I and II in alggeom/9711032 and 9712033. Here we classify maximal hyperbolic root systems of the rank three having restricted arithmetic type and a generalized lattice Weyl vector ρ with ρ 2 < 0 (i. e. of the hyperbolic type). We give classification of all reflective of hyperbolic type elementary hyperbolic lattices of the rank three. For elliptic (when ρ 2> 0) and parabolic (when ρ 2 = 0) types it was done in Parts I and II. We apply the same arguments as for elliptic and parabolic types: the method of narrow parts of polyhedra in hyperbolic spaces, and class numbers of central symmetries. But we should say that for the hyperbolic type all considerations are much more complicated and required much more calculations and time. These results are important, for example, for Theory of Lorentzian Kac–Moody algebras and some aspects of Mirror Symmetry. We also apply these results to prove boundedness of families of algebraic surfaces with almost finite polyhedral Mori cone (see math.AG/9806047 about this subject).
K3 surfaces with interesting groups of automorphisms
, 1997
"... By the fundamental result of I.I. PiatetskyShapiro and I.R. Shafarevich (1971), the automorphism group Aut(X) of a K3 surface X over C and its action on the Picard lattice SX are prescribed by the Picard lattice SX. We use this result and our method (1980) to show finiteness of the set of Picard l ..."
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Cited by 5 (5 self)
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By the fundamental result of I.I. PiatetskyShapiro and I.R. Shafarevich (1971), the automorphism group Aut(X) of a K3 surface X over C and its action on the Picard lattice SX are prescribed by the Picard lattice SX. We use this result and our method (1980) to show finiteness of the set of Picard lattices SX of rank ≥ 3 such that the automorphism group Aut(X) of the K3 surface X has a nontrivial invariant sublattice S0 in SX where the group Aut(X) acts as a finite group. For hyperbolic and parabolic lattices S0 it has been proved by the author before (1980, 1995). Thus we extend this results to negative sublattices S0. We give several examples of Picard lattices SX with parabolic and negative S0. We also formulate the corresponding finiteness result for reflective hyperbolic lattices of hyperbolic type over purely real algebraic number fields. These results are important for the theory of Lorentzian Kac–Moody algebras and Mirror Symmetry.
A LECTURE ON ARITHMETIC MIRROR SYMMETRY AND CALABIYAU MANIFOLDS
, 1996
"... Abstract. In this lecture we extend our variant of Mirror Symmetry for K3 surfaces [GN3] and clarify its relation with Mirror Symmetry for Calabi–Yau manifolds. We introduce two classes (for the models A and B) of Calabi–Yau manifolds fibrated by K3 surfaces with some special Picard lattices. These ..."
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Cited by 2 (2 self)
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Abstract. In this lecture we extend our variant of Mirror Symmetry for K3 surfaces [GN3] and clarify its relation with Mirror Symmetry for Calabi–Yau manifolds. We introduce two classes (for the models A and B) of Calabi–Yau manifolds fibrated by K3 surfaces with some special Picard lattices. These two classes are closely related with automorphic forms on IV type domains which we studied in our papers [GN1]– [GN6]. Conjecturally these automorphic forms take part in quantum intersection pairing for model A, Yukawa coupling for model B and Mirror Symmetry between these two classes of Calabi–Yau manifolds. Recently there were several papers by physicists where it was shown on some examples. We propose a problem of classification of introduced Calabi–Yau manifolds. Our papers [GN1]–[GN6] and [N4] —[N12] give a hope that this is possible. This lecture was given by the second author at the conference “Mirror Symmetry and Calabi–Yau manifolds ” held in RIMS, Kyoto on 1 December, 1996. It contains results by both authors. In [GN3] we suggested a variant of mirror symmetry for K3 surfaces which is closely related with reflection groups in hyperbolic spaces and automorphic forms on IV type domains. This subject was developed in our pares [GN1]—[GN6], [N11]. We should say that some results of R. Borcherds [B1]—[B7] are also directly related with this subject. Recently several papers by physicists have appeared where our automorphic forms (and some automorphic forms constructed by R. Borcherds) were used in Mirror Symmetry for Calabi–Yau manifolds. Physicists have shown that automorphic forms on IV type domains which we used for our variant of Mirror Symmetry for K3 surfaces [GN3] take part in the quantum intersection pairing and the Yukawa coupling for some Calabi–Yau manifolds. We only mention papers which are directly
Realizations of the Monster Lie algebra
 Selecta Math. (NS
, 1995
"... In this paper we reinterpret the theory of generalized KacMoody Lie algebras in terms of local Lie algebras formed from reductive or KacMoody algebras and certain modules (possibly infinitedimensional) for these algebras. We exploit the fact, established in [19], that certain generalized KacMood ..."
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Cited by 2 (0 self)
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In this paper we reinterpret the theory of generalized KacMoody Lie algebras in terms of local Lie algebras formed from reductive or KacMoody algebras and certain modules (possibly infinitedimensional) for these algebras. We exploit the fact, established in [19], that certain generalized KacMoody algebras
A theory of Lorentzian Kac–Moody algebras
, 1998
"... We present a variant of the Theory of Lorentzian (i. e. with a hyperbolic generalized Cartan matrix) Kac–Moody algebras recently developed by V. A. Gritsenko and the author. It is closely related with and strongly uses results of R. Borcherds. This theory should generalize wellknown Theories of fi ..."
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Cited by 1 (1 self)
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We present a variant of the Theory of Lorentzian (i. e. with a hyperbolic generalized Cartan matrix) Kac–Moody algebras recently developed by V. A. Gritsenko and the author. It is closely related with and strongly uses results of R. Borcherds. This theory should generalize wellknown Theories of finite Kac–Moody algebras (i. e. classical semisimple Lie algebras corresponding to positive generalized Cartan matrices) and affine Kac–Moody algebras (corresponding to semipositive generalized Cartan matrices). Main features of the Theory of Lorentzian Kac–Moody algebras are: One should consider generalized Kac–Moody algebras introduced by Borcherds. Denominator function should be an automorphic form on IV type Hermitian symmetric domain (first example of this type related with Leech lattice was found by Borcherds). The Kac–Moody algebra is graded by elements of an integral hyperbolic lattice S. Weyl group acts in the hyperbolic space related with S and has a fundamental polyhedron M of finite (or almost finite) volume and a lattice Weyl vector. There are results and conjectures which permit (in principle) to get a “finite” list
The Classification of the Finite Simple Groups: An Overview
 MONOGRAFÍAS DE LA REAL ACADEMIA DE CIENCIAS DE ZARAGOZA. 26: 89–104, (2004)
, 2004
"... ..."
unknown title
, 2010
"... A monster tale: a review on Borcherds ’ proof of monstrous moonshine conjecture by ..."
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A monster tale: a review on Borcherds ’ proof of monstrous moonshine conjecture by